Finite Group Databasebeta

← #16023387

Subgroups — #16023387

Order 3328.

2422 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88198C211yesmaximal, characteristic
2#13117924C214yesmaximal, complemented
3#13117893C212yesmaximal, complemented
4#13117890C212yesmaximal, complemented
5#13117714C212yesmaximal
6#13117708C212yesmaximal
7Q8×C22⋊Dic13C211yesmaximal, characteristic
8C22⋊C4×Dic26C211yesmaximal, characteristic
9#13117668C211yesmaximal, characteristic
10#13117579C214yesmaximal, complemented
11#13117453C212yesmaximal
12#13117405C212yesmaximal, complemented
13C4×#88198C212yesmaximal, complemented
14#13117238C211yesmaximal, characteristic
15#13117191C212yesmaximal
16C4×#88131C211yesmaximal, characteristic
17Q8×C13×C22⋊C4C211yesmaximal, characteristic
18C23×Q8×C13C2211yescharacteristic
19C23×Dic26C2211yescharacteristic
20C2×C4×Q8×C13C2212yes—
21C2×Q8×Dic13C2212yes—
22C2×#88198C4116yescomplemented
23C2×#88198C2212yes—
24C2×#88198C2211yescharacteristic
25C2×#88198C2211yescharacteristic
26C2×#88198C2211yescharacteristic
27C2×#88198C2211yescharacteristic
28C2×#88198C2212yes—
29C22×#6101C2212yes—
30C22×#6101C2212yes—
31C22×C4×Dic13C2211yescharacteristic
32C2×#88141C2212yes—
33C2×#88131C2212yes—
34C2×#88131C2211yescharacteristic
35C2×#88131C2211yescharacteristic
36C2×C4×Dic26C2212yes—
37C13×#391C2214yes—
38C13×#391C2212yes—
39C13×#380C2212yes—
40C13×#380C2211yescharacteristic
41C4×C13×C22⋊C4C2212yes—
42C4×C13×C22⋊C4C2211yescharacteristic
43#11718891C2214yes—
44#11718891C2212yes—
45#11718873C2212yes—
46#11718873C2211yescharacteristic
47C4×C22⋊Dic13C2212yes—
48C4×C22⋊Dic13C2211yescharacteristic
49#11718639C2214yes—
50#11718637C2212yes—
51C4⋊C4×Dic13C2212yes—
52#11718635C2212yes—
53#11718633C2214yes—
54#11718633C2214yes—
55#11718633C2214yes—
56#11718603C2212yes—
57#11718603C2212yes—
58#11718603C2212yes—
59C22⋊C4×Dic13C2212yes—
60C22⋊C4×Dic13C2212yes—
61C22⋊C4×Dic13C2211yescharacteristic
62C22⋊C4×Dic13C2211yescharacteristic
63#11718594C2214yes—
64C4×#6101C2214yes—
65C4×#6101C2214yes—
66C4×#6101C2214yes—
67C4×#6101C2212yes—
68C42×Dic13C2212yes—
69#11718590C2212yes—
70#11718316C2214yes—
71#11718313C2214yes—
72#11718311C2212yes—
73#11718310C2214yes—
74#11718310C2214yes—
75#11718310C2214yes—
76#11718310C2212yes—
77#11718310C2212yes—
78#11718309C2212yes—
79#11718308C2214yes—
80#11718308C2214yes—
81#11718308C2214yes—
82#11718307C2214yes—
83#11718307C2214yes—
84C22×Q8×C13—44no—
85C22×Q8×C13C2×C418yes—
86C22×Q8×C13C2311yescharacteristic
87C22×Q8×C13C2311yescharacteristic
88C22×Q8×C13C2312yes—
89C23×C4×C13C2311yescharacteristic
90C23×C4×C13C2312yes—
91C23×Dic13C2311yescharacteristic
92C23×Dic13C2312yes—
93C23×Dic13C2311yescharacteristic
94C22×Dic26—44no—
95C22×Dic26C2×C418yes—
96C22×Dic26C2311yescharacteristic
97C22×Dic26C2311yescharacteristic
98C22×Dic26C2312yes—
99C4×Q8×C13—44no—
100C2×C13×C4⋊C4C2312yes—
101C2×C13×C4⋊C4C2314yes—
102C2×C13×C22⋊C4C2312yes—
103C2×C13×C22⋊C4C2311yescharacteristic
104C2×C13×C22⋊C4C2311yescharacteristic
105C2×C42×C13C2312yes—
106C2×C42×C13C2314yes—
107C2×C22⋊Dic13C2312yes—
108C2×C22⋊Dic13C2311yescharacteristic
109C2×C22⋊Dic13C2311yescharacteristic
110Q8×Dic13—44no—
111#88198C2×C418yes—
112#88198C2×C418yes—
113#88198C2×C418yes—
114#88198C2×C418yes—
115#88198C2×C4116yes—
116#88198C2×C4116yes—
117C2×C4⋊Dic13C2312yes—
118C2×C4⋊Dic13C2314yes—
119C2×#6101C2×C4116yes—
120C2×#6101C2×C4116yes—
121C2×#6101C2312yes—
122C2×#6101C2314yes—
123C2×#6101C2314yes—
124C2×#6101C2314yes—
125C2×#6101C2314yes—
126C2×#6101C2312yes—
127C2×#6101C2312yes—
128C2×#6101C2312yes—
129C2×#6101C2312yes—
130C2×#6101C2312yes—
131C2×#6101C2312yes—
132C2×#6101C2312yes—
133C2×#6101C2312yes—
134C2×#6101C2312yes—
135C2×C4×Dic13C2×C418yes—
136C2×C4×Dic13C2312yes—
137C2×C4×Dic13C2314yes—
138C2×C4×Dic13C2312yes—
139C2×C4×Dic13C2312yes—
140C2×C4×Dic13C2312yes—
141C2×C4×Dic13C2314yes—
142C2×C4×Dic13C2314yes—
143C2×C4×Dic13C2314yes—
144C2×C4×Dic13C2311yescharacteristic
145C2×C4×Dic13C2311yescharacteristic
146C2×C4×Dic13C2311yescharacteristic
147C2×C4×Dic13C2311yescharacteristic
148C2×C4×Dic13C2312yes—
149#88141—44no—
150#88131—28no—
151#88131—24no—
152#88131Q818yescomplemented
153C4×Dic26—44no—
154C13×#95C2312yes—
155C13×#95C2314yes—
156C13×#95C2312yes—
157C13×#95C2314yes—
158#88071C2312yes—
159#88071C2312yes—
160#88071C2314yes—
161#88071C2314yes—
162#88071C2314yes—
163#88071C2314yes—
164#88071C2312yes—
165#88071C2312yes—
166#88071C2314yes—
167#88071C2314yes—
168#88071C2314yes—
169#34216—131nomaximal, Sylow
170C24×C13C2411yescharacteristic
171C2×Q8×C13—24no—
172C2×Q8×C13—24no—
173C2×Q8×C13—44no—
174C2×Q8×C13—416no—
175C2×Q8×C13C22×C414yes—
176C2×Q8×C13C22×C414yes—
177C2×Q8×C13C22×C418yes—
178C22×C4×C13—24no—
179C22×C4×C13—44no—
180C22×C4×C13C22×C414yes—
181C22×C4×C13C22×C418yes—
182C22×C4×C13C2414yes—
183C22×C4×C13C2412yes—
184C22×C4×C13C2411yescharacteristic
185C22×C4×C13C2412yes—
186C22×C4×C13C2412yes—
187C22×C4×C13C2412yes—
188C22×C4×C13C2411yescharacteristic
189C22×Dic13—24no—
190C22×Dic13—44no—
191C22×Dic13C22×C414yes—
192C22×Dic13C22×C418yes—
193C22×Dic13C22×C414yes—
194C22×Dic13C2×Q814yes—
195C22×Dic13#4114yes—
196C22×Dic13C2412yes—
197C22×Dic13C2414yes—
198C22×Dic13C2411yescharacteristic
199C22×Dic13C2412yes—
200C22×Dic13C2412yes—
201C22×Dic13C2412yes—
202C22×Dic13C2411yescharacteristic
203C22×Dic13C2411yescharacteristic
204C22×Dic13C2411yescharacteristic
205C2×Dic26—24no—
206C2×Dic26—24no—
207C2×Dic26—416no—
208C2×Dic26—44no—
209C2×Dic26C22×C414yes—
210C2×Dic26C22×C414yes—
211C2×Dic26C22×C418yes—
212C13×C4⋊C4—24no—
213C13×C4⋊C4—44no—
214C13×C22⋊C4—24no—
215C13×C22⋊C4—22no—
216C13×C22⋊C4C2×Q814yes—
217C42×C13—24no—
218C42×C13—44no—
219C22⋊Dic13—24no—
220C22⋊Dic13—22no—
221C22⋊Dic13C2×Q814yes—
222C4⋊Dic13—24no—
223C4⋊Dic13—44no—
224#6101—24no—
225#6101—28no—
226#6101—28no—
227#6101—44no—
228#6101—44no—
229#6101C22×C418yes—
230#6101C22×C418yes—
231#6101C22×C418yes—
232#6101C22×C418yes—
233#6101C22×C418yes—
234#6101C22×C418yes—
235#6101C22×C418yes—
236#6101C22×C418yes—
237#6101C2×Q818yescomplemented
238C4×Dic13—24no—
239C4×Dic13—28no—
240C4×Dic13—24no—
241C4×Dic13—24no—
242C4×Dic13—24no—
243C4×Dic13—44no—
244C4×Dic13—44no—
245C4×Dic13—44no—
246C4×Dic13C22×C414yes—
247C4×Dic13C22×C414yes—
248C4×Dic13C22×C414yes—
249C4×Dic13C22×C414yes—
250C4×Dic13C2×Q818yescomplemented
251C22×C4⋊Q8—131no—
252#2369—134no—
253#2365—132no—
254#2365—132no—
255#2345—132no—
256#2345—132no—
257Q8×C22⋊C4—131no—
258Q8×C22⋊C4—131no—
259Q8×C22⋊C4—131no—
260Q8×C22⋊C4D1311yescharacteristic
261#2331—134no—
262#2316—132no—
263C4×C4⋊Q8—132no—
264C4×C4⋊Q8—132no—
265#2285—131no—
266#2274—132no—
267C4×#117—131no—
268C23×C13—22no—
269C23×C13—22no—
270C23×C13C23×C412yes—
271C23×C13C22×Q812yes—
272C23×C13C2×#4112yes—
273C23×C13C2511yescharacteristic
274Q8×C13—28no—
275Q8×C13—28no—
276Q8×C13—48no—
277C2×C4×C13—22no—
278C2×C4×C13—24no—
279C2×C4×C13—24no—
280C2×C4×C13—22no—
281C2×C4×C13—22no—
282C2×C4×C13—28no—
283C2×C4×C13—24no—
284C2×C4×C13—22no—
285C2×C4×C13—24no—
286C2×C4×C13—22no—
287C2×C4×C13—22no—
288C2×C4×C13—24no—
289C2×C4×C13—48no—
290C2×C4×C13C23×C412yes—
291C2×C4×C13C23×C414yes—
292C2×C4×C13C23×C414yes—
293C2×C4×C13C23×C412yes—
294C2×C4×C13C22×Q814yes—
295C2×C4×C13H(5,𝔽2)14yes—
296C2×Dic13—22no—
297C2×Dic13—24no—
298C2×Dic13—24no—
299C2×Dic13—22no—
300C2×Dic13—22no—
301C2×Dic13—28no—
302C2×Dic13—22no—
303C2×Dic13—24no—
304C2×Dic13—22no—
305C2×Dic13—24no—
306C2×Dic13—22no—
307C2×Dic13—24no—
308C2×Dic13—48no—
309C2×Dic13C4×Q8116yescomplemented
310C2×Dic13C23×C412yes—
311C2×Dic13C23×C414yes—
312C2×Dic13C23×C414yes—
313C2×Dic13C23×C412yes—
314C2×Dic13C23×C412yes—
315C2×Dic13C23×C412yes—
316C2×Dic13C22×Q814yes—
317C2×Dic13C22×Q812yes—
318C2×Dic13C22×Q812yes—
319C2×Dic13C2×#4112yes—
320C2×Dic13C2×#4112yes—
321C2×Dic132−1+414yes—
322Dic26—28no—
323Dic26—28no—
324Dic26—48no—
325C23×Q8—131no—
326C23×Q8C2×D1311yescharacteristic
327C2×C4⋊Q8—1316no—
328C2×C4⋊Q8—131no—
329C2×C4⋊Q8—131no—
330C2×C4⋊Q8—132no—
331C2×C4⋊Q8—131no—
332C2×C4⋊Q8—131no—
333C2×C4⋊Q8—132no—
334C2×C4×Q8—132no—
335C2×C4×Q8—132no—
336C2×C4×Q8—132no—
337C2×C4×Q8C2×D1312yes—
338C2×#117—131no—
339C2×#117—131no—
340C2×#117—132no—
341C22×C4⋊C4—132no—
342C22×C4⋊C4—132no—
343C22×C42—131no—
344#391—132no—
345#391—134no—
346#391—132no—
347#391—132no—
348#391—134no—
349#391—134no—
350#391C2×D1312yes—
351#391C2×D1314yes—
352#389—134no—
353#389—134no—
354#389—134no—
355#389—134no—
356#389—134no—
357#389—134no—
358#387—134no—
359#387—134no—
360#387—134no—
361#387—134no—
362#384—132no—
363#382—132no—
364#380—132no—
365#380—131no—
366#380—132no—
367#380—132no—
368#380—132no—
369#380C2×D1312yes—
370#380C2×D1311yescharacteristic
371C4×C4⋊C4—132no—
372C4×C4⋊C4—134no—
373C4×C4⋊C4—134no—
374C4×C4⋊C4—132no—
375C4×C4⋊C4—134no—
376C4×C4⋊C4—134no—
377C4×C4⋊C4—134no—
378C4×C4⋊C4—134no—
379C4×C4⋊C4—132no—
380C4×C4⋊C4—132no—
381C4×C22⋊C4—131no—
382C4×C22⋊C4—132no—
383C4×C22⋊C4—132no—
384C4×C22⋊C4—132no—
385C4×C22⋊C4—131no—
386C4×C22⋊C4—131no—
387C4×C22⋊C4C2×D1311yescharacteristic
388C4×C22⋊C4C2×D1312yes—
389#376—132no—
390C43—132no—
391C22×C13—22no—
392C22×C13—22no—
393C22×C13—24no—
394C22×C13—24no—
395C22×C13C2×C4×Q814yes—
396C22×C13C24×C411yescharacteristic
397C22×C13C23×Q811yescharacteristic
398C22×C13C22×#4111yescharacteristic
399C22×C13C2×H(5,𝔽2)12yes—
400C22×C13C2×2−1+412yes—
401C4×C13—22no—
402C4×C13—24no—
403C4×C13—24no—
404C4×C13—24no—
405C4×C13—44no—
406C4×C13—42no—
407C4×C13#51814yes—
408Dic13—24no—
409Dic13—24no—
410Dic13—22no—
411Dic13—24no—
412Dic13—42no—
413Dic13—44no—
414Dic13C2×C4×Q814yes—
415Dic13C2×C4×Q814yes—
416Dic13#51914yes—
417C22×Q8—44no—
418C22×Q8—138no—
419C22×Q8—131no—
420C22×Q8—131no—
421C22×Q8—132no—
422C22×Q8—524no—
423C22×Q8C4×D1318yes—
424C22×Q8C22×D1311yescharacteristic
425C22×Q8C22×D1311yescharacteristic
426C22×Q8C22×D1312yes—
427C23×C4—131no—
428C23×C4—131no—
429C23×C4—132no—
430C23×C4C22×D1311yescharacteristic
431C23×C4C22×D1312yes—
432C4⋊Q8—138no—
433C4⋊Q8—138no—
434C4⋊Q8—138no—
435C4⋊Q8—138no—
436C4⋊Q8—1316no—
437C4⋊Q8—1316no—
438C4×Q8—44no—
439C4×Q8—524no—
440C4×Q8—524no—
441C4×Q8—524no—
442#117—138no—
443#117—264no—
444#117—268no—
445C2×C4⋊C4—1316no—
446C2×C4⋊C4—1316no—
447C2×C4⋊C4—132no—
448C2×C4⋊C4—132no—
449C2×C4⋊C4—132no—
450C2×C4⋊C4—132no—
451C2×C4⋊C4—132no—
452C2×C4⋊C4—132no—
453C2×C4⋊C4—132no—
454C2×C4⋊C4—132no—
455C2×C4⋊C4—132no—
456C2×C4⋊C4—134no—
457C2×C4⋊C4—132no—
458C2×C4⋊C4—132no—
459C2×C4⋊C4—134no—
460C2×C4⋊C4—134no—
461C2×C4⋊C4—134no—
462C2×C4⋊C4—134no—
463C2×C4⋊C4C22×D1312yes—
464C2×C4⋊C4C22×D1314yes—
465C2×C22⋊C4—131no—
466C2×C22⋊C4—131no—
467C2×C22⋊C4—132no—
468C2×C22⋊C4C22×D1311yescharacteristic
469C2×C22⋊C4C22×D1312yes—
470C2×C22⋊C4C22×D1311yescharacteristic
471C2×C42—138no—
472C2×C42—132no—
473C2×C42—131no—
474C2×C42—131no—
475C2×C42—131no—
476C2×C42—131no—
477C2×C42—132no—
478C2×C42—134no—
479C2×C42—132no—
480C2×C42—132no—
481C2×C42—132no—
482C2×C42—134no—
483C2×C42—134no—
484C2×C42—134no—
485C2×C42C22×D1312yes—
486C2×C42C22×D1314yes—
487#95—132no—
488#95—132no—
489#95—132no—
490#95—134no—
491#95—134no—
492#95—134no—
493#95—134no—
494#95—132no—
495#95—134no—
496#95—134no—
497#95—134no—
498#95C22×D1312yes—
499#95C22×D1314yes—
500#95C22×D1312yes—
501#95C22×D1314yes—
502C2×C13—24no—
503C2×C13C22×C4×Q811yescharacteristic
504C2×C13C2×#51811yescharacteristic
505C2×C13C2×#51911yescharacteristic
506C2×C13#350312yes—
507C2×C13#351112yes—
508C24C23×D1311yescharacteristic
509C2×Q8—24no—
510C2×Q8—24no—
511C2×Q8—44no—
512C2×Q8—416no—
513C2×Q8—134no—
514C2×Q8—134no—
515C2×Q8—138no—
516C2×Q8—264no—
517C2×Q8—264no—
518C2×Q8—524no—
519C2×Q8—5216no—
520C2×Q8C2×C4×D1314yes—
521C2×Q8C2×C4×D1314yes—
522C2×Q8C2×C4×D1318yes—
523C22×C4—24no—
524C22×C4—44no—
525C22×C4—134no—
526C22×C4—134no—
527C22×C4—134no—
528C22×C4—131no—
529C22×C4—131no—
530C22×C4—131no—
531C22×C4—134no—
532C22×C4—131no—
533C22×C4—132no—
534C22×C4—132no—
535C22×C4—132no—
536C22×C4—138no—
537C22×C4—132no—
538C22×C4—134no—
539C22×C4—264no—
540C22×C4—524no—
541C22×C4C2×C4×D1314yes—
542C22×C4C2×C4×D1318yes—
543C22×C4C23×D1312yes—
544C22×C4C23×D1314yes—
545C22×C4C23×D1311yescharacteristic
546C22×C4C23×D1312yes—
547C22×C4C23×D1312yes—
548C22×C4C23×D1312yes—
549C22×C4C23×D1311yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—44no—
552C4⋊C4—138no—
553C4⋊C4—138no—
554C4⋊C4—138no—
555C4⋊C4—138no—
556C4⋊C4—138no—
557C4⋊C4—138no—
558C4⋊C4—138no—
559C4⋊C4—138no—
560C4⋊C4—138no—
561C4⋊C4—264no—
562C4⋊C4—268no—
563C4⋊C4—268no—
564C4⋊C4—264no—
565C4⋊C4—524no—
566C4⋊C4—524no—
567C4⋊C4—524no—
568C22⋊C4—22no—
569C22⋊C4—24no—
570C22⋊C4—134no—
571C22⋊C4—262no—
572C22⋊C4—264no—
573C22⋊C4Q8×D1314yes—
574C42—24no—
575C42—44no—
576C42—134no—
577C42—134no—
578C42—134no—
579C42—134no—
580C42—138no—
581C42—268no—
582C42—264no—
583C42—264no—
584C42—264no—
585C42—264no—
586C42—524no—
587C42—524no—
588C42—524no—
589C13#3421611yescharacteristic, complemented, Sylow
590C23—22no—
591C23—22no—
592C23C22×C4×D1312yes—
593C23C2×Q8×D1312yes—
594C23#41×D1312yes—
595C23C24×D1311yescharacteristic
596Q8—28no—
597Q8—28no—
598Q8—48no—
599Q8—268no—
600Q8—268no—
601Q8—528no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—48no—
615C2×C4—1316no—
616C2×C4—132no—
617C2×C4—132no—
618C2×C4—132no—
619C2×C4—132no—
620C2×C4—132no—
621C2×C4—132no—
622C2×C4—132no—
623C2×C4—134no—
624C2×C4—132no—
625C2×C4—134no—
626C2×C4—134no—
627C2×C4—134no—
628C2×C4—268no—
629C2×C4—262no—
630C2×C4—264no—
631C2×C4—264no—
632C2×C4—262no—
633C2×C4—262no—
634C2×C4—262no—
635C2×C4—264no—
636C2×C4—262no—
637C2×C4—264no—
638C2×C4—262no—
639C2×C4—264no—
640C2×C4—528no—
641C2×C4C22×C4×D1312yes—
642C2×C4C22×C4×D1314yes—
643C2×C4C22×C4×D1314yes—
644C2×C4C22×C4×D1312yes—
645C2×C4#8825114yes—
646C2×C4C2×Q8×D1314yes—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22C4×Q8×D1314yes—
652C22C23×C4×D1311yescharacteristic
653C22C2×#8825112yes—
654C22C22×Q8×D1311yescharacteristic
655C22C2×#41×D1311yescharacteristic
656C22D13×2−1+412yes—
657C4—22no—
658C4—24no—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—134no—
664C4—134no—
665C4—134no—
666C4—264no—
667C4—264no—
668C4—262no—
669C4—264no—
670C4—522no—
671C4—524no—
672C4#1171919114yes—
673C2—24no—
674C2C2×C4×Q8×D1311yescharacteristic
675C2C2×#1171919111yescharacteristic
676C2D13×#51911yescharacteristic
677C2#1312864712yes—
678C2#1312874212yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingQ8×C13×C22⋊C4C2
socle, cosocleC23×C13C25
prime radicalC24×C13C24
hypercenterQ8×C22⋊C4D13
hypocenterC13#34216
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13117924C2semidirect
3. #13117893C2semidirect
4. #13117890C2semidirect
10. #13117579C2semidirect
12. #13117405C2semidirect
13. C4×#88198C2semidirect
22. C2×#88198C4semidirect
152. #88131Q8semidirect
237. #6101C2×Q8semidirect
250. C4×Dic13C2×Q8semidirect
309. C2×Dic13C4×Q8semidirect
589. C13#34216semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#881982
#131179242
#131178932
#131178902
#131177142
#131177082
Q8×C22⋊Dic132
C22⋊C4×Dic262
#131176682
#131175792
#131174532
#131174052
C4×#881982
#131172382
#131171912
C4×#881312
Q8×C13×C22⋊C42
#3421613