Finite Group Databasebeta

← #10981905

Subgroups — #10981905

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic6C211yesmaximal, characteristic, complemented
2C2×#5516C211yesmaximal, characteristic
3C3×#2396C211yesmaximal, characteristic
4#82480C211yesmaximal, characteristic, complemented
5#82264C211yesmaximal, characteristic
6#78181C212yesmaximal, complemented
7#78049C212yesmaximal
8#77748C214yesmaximal, complemented
9#77288C214yesmaximal
10#77191C212yesmaximal, complemented
11#76778C212yesmaximal
12#73722C211yesmaximal, characteristic, complemented
13#73674C212yesmaximal, complemented
14#73637C211yesmaximal, characteristic, complemented
15#73637C212yesmaximal, complemented
16#73492C212yesmaximal
17C22⋊C4×Dic6C212yesmaximal
18#35711—31nomaximal, Sylow
19C23×Dic6C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×Q8×Dic3C2212yes—
23C2×#926C2211yescharacteristic
24Q8×Dic6—28no—
25#5516—28no—
26C2×#888C2212yes—
27C2×#888C2214yes—
28C2×#887C2212yes—
29C2×#887C2212yes—
30C2×#877C2212yes—
31C2×#877C2212yes—
32C2×#869C2212yes—
33C2×#869C2211yescharacteristic
34C2×C4×Dic6C2212yes—
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37C2×C4×Dic6C2212yes—
38C2×C4×Dic6C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40C3×#400C2211yescharacteristic
41C3×#399C2211yescharacteristic
42C3×#398C2212yes—
43C3×#396C2212yes—
44C3×#395C2211yescharacteristic
45C3×#393C2211yescharacteristic
46C3×C4×C4⋊C4C2212yes—
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C22⋊C4C2212yes—
49#5230C2212yes—
50#5212C2212yes—
51#4986C2212yes—
52#4985C2212yes—
53#4980C2214yes—
54#4979C2212yes—
55#4979C2214yes—
56#4978C2214yes—
57#4976C2214yes—
58#4976C2214yes—
59#4976C2214yes—
60#4973C2214yes—
61#4949C2212yes—
62#4948C2214yes—
63#4947C2214yes—
64#4947C2212yes—
65#4942C2214yes—
66Dic3×C22⋊C4C2214yes—
67#4936C2212yes—
68#4936C2211yescharacteristic
69#4929C2212yes—
70#4929C2212yes—
71#4928C2212yes—
72#4928C2212yes—
73#4928C2212yes—
74#4659C2212yes—
75#4659C2212yes—
76#4657C2214yes—
77#4655C2214yes—
78#4650C2214yes—
79#4650C2212yes—
80#4649C2212yes—
81#4649C2214yes—
82#4646C2214yes—
83#4646C2214yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×Q82—31no—
87C2×#519—31no—
88#2628—32no—
89#2628—31no—
90#2581—32no—
91#2581—31no—
92#2568—32no—
93#2568—31no—
94#2451—34no—
95#2451—32no—
96#2439—34no—
97#2439—32no—
98#2396—32no—
99#2396D311yescharacteristic
100#2351—31no—
101#2351—32no—
102Q8×C22⋊C4—32no—
103C22×C3×Q8C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×Dic3C2312yes—
106C22×Dic6—24no—
107C22×Dic6—24no—
108C22×Dic6Q818yescomplemented
109C22×Dic6C2312yes—
110C22×Dic6C2311yescharacteristic
111C22×Dic6C2314yes—
112C22×Dic6C2312yes—
113C22×Dic6C2311yescharacteristic
114C3×C4×Q8—24no—
115C3×#117—24no—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C22⋊C4C2312yes—
124C2×C3×C22⋊C4C2311yescharacteristic
125C2×C3×C22⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4C2312yes—
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2312yes—
129C2×C3×C42C2312yes—
130C2×C3×C42C2311yescharacteristic
131C2×#217C2312yes—
132C2×#217C2314yes—
133Q8×Dic3—28no—
134#926—24no—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2312yes—
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2314yes—
153C2×C4×Dic3C2314yes—
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2312yes—
157#888—216no—
158#888—28no—
159#887—28no—
160#887—28no—
161#877—28no—
162#877—28no—
163#869—24no—
164#869—28no—
165C4×Dic6—24no—
166C4×Dic6—28no—
167C4×Dic6—28no—
168C4×Dic6—24no—
169C4×Dic6—24no—
170C4×Dic6D418yes—
171C3×#95C2312yes—
172C3×#95C2312yes—
173C3×#95C2312yes—
174C3×#95C2312yes—
175C3×#95C2311yescharacteristic
176C3×#95C2311yescharacteristic
177C3⋊#95C2314yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2312yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C23×Q8—31no—
190Q82—68no—
191C2×C4⋊Q8—32no—
192C2×C4⋊Q8—34no—
193C2×C4⋊Q8—31no—
194C2×C4⋊Q8—32no—
195#519—68no—
196C2×C4×Q8—32no—
197C2×C4×Q8—32no—
198C2×C4×Q8—32no—
199C2×C4×Q8—31no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—32no—
203C2×C4×Q8—31no—
204C2×C4×Q8—31no—
205C2×C4×Q8C2×D311yescharacteristic
206C2×#117—31no—
207C2×#117—32no—
208C2×#117—32no—
209C2×#117C2×D311yescharacteristic
210#400—32no—
211#400—32no—
212#400—32no—
213#400—32no—
214#400C2×D311yescharacteristic
215#399—32no—
216#399C2×D311yescharacteristic
217#398—34no—
218#398C2×D312yes—
219#396—34no—
220#396C2×D312yes—
221#395—32no—
222#395—34no—
223#395—32no—
224#395C2×D311yescharacteristic
225#393—34no—
226#393—32no—
227#393C2×D311yescharacteristic
228#391—34no—
229#391—32no—
230#391—32no—
231#391—34no—
232#391—34no—
233#391—34no—
234#391—34no—
235#389—34no—
236#389—34no—
237#389—34no—
238#389—32no—
239#389—32no—
240#389—32no—
241#387—34no—
242#387—34no—
243#387—34no—
244#382—32no—
245#382—31no—
246#382—34no—
247#382—32no—
248#380—32no—
249#380—34no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—34no—
252C4×C4⋊C4C2×D311yescharacteristic
253C4×C4⋊C4C2×D312yes—
254C4×C22⋊C4—34no—
255C4×C22⋊C4C2×D312yes—
256C24×C3C2411yescharacteristic
257C2×C3×Q8—22no—
258C2×C3×Q8—24no—
259C22×C3×C4—22no—
260C22×C3×C4—22no—
261C22×C3×C4C2×Q814yes—
262C22×C3×C4C2412yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2411yescharacteristic
265C22×C3×C4C2412yes—
266C22×C3×C4C2411yescharacteristic
267C22×C3×C4C2411yescharacteristic
268C22×C3×C4C2411yescharacteristic
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2412yes—
271C22×C3×C4C2411yescharacteristic
272C22×Dic3—24no—
273C22×Dic3—24no—
274C22×Dic3C2×Q818yes—
275C22×Dic3C2414yes—
276C22×Dic3C2412yes—
277C22×Dic3C2414yes—
278C22×Dic3C2412yes—
279C22×Dic3C2412yes—
280C22×Dic3C2412yes—
281C2×Dic6—22no—
282C2×Dic6—22no—
283C2×Dic6—24no—
284C2×Dic6—24no—
285C2×Dic6—232no—
286C2×Dic6—24no—
287C2×Dic6—28no—
288C2×Dic6—44no—
289C2×Dic6—44no—
290C2×Dic6—44no—
291C2×Dic6C2×D414yes—
292C2×Dic6C2×Q814yes—
293C2×Dic6C2×Q818yes—
294C2×Dic6C2×Q814yes—
295C2×Dic6#4114yes—
296C3×C4⋊C4—22no—
297C3×C4⋊C4—22no—
298C3×C4⋊C4—24no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—24no—
301C3×C4⋊C4—42no—
302C3×C4⋊C4—42no—
303C3×C22⋊C4—22no—
304C3×C22⋊C4—22no—
305C3×C22⋊C4—24no—
306C3×C22⋊C4—42no—
307C3×C42—22no—
308C3×C42—24no—
309C3×C42—24no—
310C3×C42C2×D414yes—
311#217—24no—
312#217—44no—
313C4⋊Dic3—22no—
314C4⋊Dic3—24no—
315C4⋊Dic3—24no—
316C4⋊Dic3—24no—
317C4⋊Dic3—28no—
318C4⋊Dic3C2×D414yes—
319Dic3⋊C4—24no—
320Dic3⋊C4—28no—
321Dic3⋊C4—24no—
322Dic3⋊C4—24no—
323Dic3⋊C4—28no—
324Dic3⋊C4—28no—
325Dic3⋊C4—44no—
326Dic3⋊C4—44no—
327Dic3⋊C4C2×D418yes—
328C4×Dic3—24no—
329C4×Dic3—24no—
330C4×Dic3—24no—
331C4×Dic3—28no—
332C4×Dic3—24no—
333C4×Dic3—28no—
334C4×Dic3C2×D418yes—
335C22×Q8—38no—
336C22×Q8—31no—
337C22×Q8—32no—
338C22×Q8—31no—
339C22×Q8—34no—
340C22×Q8—32no—
341C22×Q8—64no—
342C22×Q8—64no—
343C22×Q8C22×D311yescharacteristic
344C23×C4—32no—
345C23×C4C22×D311yescharacteristic
346C4⋊Q8—68no—
347C4⋊Q8—68no—
348C4⋊Q8—616no—
349C4⋊Q8—64no—
350C4×Q8—24no—
351C4×Q8—38no—
352C4×Q8—64no—
353C4×Q8—64no—
354C4×Q8—68no—
355C4×Q8—68no—
356C4×Q8—68no—
357C4×Q8—68no—
358C4×Q8—68no—
359C4×Q8—64no—
360#117—24no—
361#117—68no—
362#117—68no—
363#117—64no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—34no—
378C2×C4⋊C4—34no—
379C2×C4⋊C4C22×D311yescharacteristic
380C2×C4⋊C4C22×D311yescharacteristic
381C2×C4⋊C4C22×D312yes—
382C2×C4⋊C4C22×D311yescharacteristic
383C2×C4⋊C4C22×D312yes—
384C2×C4⋊C4C22×D312yes—
385C2×C4⋊C4C22×D312yes—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—34no—
388C2×C22⋊C4C22×D311yescharacteristic
389C2×C22⋊C4C22×D311yescharacteristic
390C2×C22⋊C4C22×D312yes—
391C2×C22⋊C4C22×D312yes—
392C2×C42—32no—
393C2×C42—32no—
394C2×C42—32no—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—34no—
398C2×C42—34no—
399C2×C42C22×D311yescharacteristic
400C2×C42C22×D312yes—
401C2×C42C22×D312yes—
402C2×C42C22×D311yescharacteristic
403#95—34no—
404#95—34no—
405#95—34no—
406#95—32no—
407#95—34no—
408#95—34no—
409#95—34no—
410#95—34no—
411#95—32no—
412#95—34no—
413#95—34no—
414#95—34no—
415#95C22×D312yes—
416#95C22×D312yes—
417#95C22×D312yes—
418#95C22×D311yescharacteristic
419#95C22×D312yes—
420#95C22×D311yescharacteristic
421C23×C3—21no—
422C23×C3—21no—
423C23×C3—22no—
424C23×C3—22no—
425C23×C3C22×Q812yes—
426C23×C3C2511yescharacteristic
427C3×Q8—28no—
428C2×C3×C4—22no—
429C2×C3×C4—21no—
430C2×C3×C4—21no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—28no—
435C2×C3×C4—21no—
436C2×C3×C4—21no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—22no—
448C2×C3×C4C22×D412yes—
449C2×C3×C4C22×D412yes—
450C2×C3×C4C22×D412yes—
451C2×C3×C4C22×Q812yes—
452C2×C3×C4C22×Q812yes—
453C2×C3×C4C2×#4112yes—
454C2×C3×C42−1+414yes—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—24no—
458C2×Dic3—216no—
459C2×Dic3—22no—
460C2×Dic3—24no—
461C2×Dic3—24no—
462C2×Dic3—24no—
463C2×Dic3—24no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3—24no—
467C2×Dic3C22×D414yes—
468C2×Dic3C22×D414yes—
469C2×Dic3C22×Q814yes—
470C2×Dic3C22×Q814yes—
471C2×Dic3C2×#4114yes—
472Dic6—28no—
473Dic6—216no—
474Dic6—88no—
475Dic6—84no—
476Dic6C22⋊Q8116yes—
477C24C23×D311yescharacteristic
478C2×Q8—22no—
479C2×Q8—24no—
480C2×Q8—34no—
481C2×Q8—34no—
482C2×Q8—38no—
483C2×Q8—34no—
484C2×Q8—34no—
485C2×Q8—64no—
486C2×Q8—64no—
487C2×Q8—68no—
488C2×Q8—632no—
489C2×Q8—62no—
490C2×Q8—64no—
491C2×Q8—62no—
492C2×Q8—124no—
493C2×Q8—124no—
494C2×Q8—124no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—32no—
498C22×C4—38no—
499C22×C4—32no—
500C22×C4—32no—
501C22×C4—34no—
502C22×C4—32no—
503C22×C4—34no—
504C22×C4—64no—
505C22×C4—64no—
506C22×C4D3×Q814yes—
507C22×C4C23×D312yes—
508C22×C4C23×D312yes—
509C22×C4C23×D311yescharacteristic
510C22×C4C23×D312yes—
511C22×C4C23×D311yescharacteristic
512C22×C4C23×D311yescharacteristic
513C22×C4C23×D311yescharacteristic
514C22×C4C23×D311yescharacteristic
515C22×C4C23×D312yes—
516C22×C4C23×D311yescharacteristic
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—38no—
523C4⋊C4—34no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—64no—
527C4⋊C4—64no—
528C4⋊C4—68no—
529C4⋊C4—68no—
530C4⋊C4—68no—
531C4⋊C4—64no—
532C4⋊C4—62no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—64no—
536C4⋊C4—68no—
537C4⋊C4—124no—
538C4⋊C4—124no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—64no—
544C22⋊C4—124no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—38no—
549C42—64no—
550C42—64no—
551C42—64no—
552C42—68no—
553C42—68no—
554C42—64no—
555C42D3×D414yes—
556C22×C3—22no—
557C22×C3—24no—
558C22×C3—41no—
559C22×C3—41no—
560C22×C3—42no—
561C22×C3C23×D411yescharacteristic
562C22×C3C23×Q811yescharacteristic
563C22×C3C22×#4111yescharacteristic
564C22×C3C2×2−1+412yes—
565C22×C3#58512yes—
566C3×C4—22no—
567C3×C4—22no—
568C3×C4—24no—
569C3×C4—22no—
570C3×C4—41no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—42no—
574C3×C4—42no—
575C3×C4C2×C22⋊Q814yes—
576Dic3—24no—
577Dic3—24no—
578Dic3—42no—
579Dic3—44no—
580Dic3—44no—
581Dic3C2×C22⋊Q818yes—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—22no—
586C23C2×D3×Q812yes—
587C23C24×D311yescharacteristic
588Q8—28no—
589Q8—316no—
590Q8—616no—
591Q8—68no—
592Q8—244no—
593Q8—248no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—34no—
615C2×C4—34no—
616C2×C4—34no—
617C2×C4—34no—
618C2×C4—34no—
619C2×C4—616no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—64no—
631C2×C4C2×D3×D412yes—
632C2×C4C2×D3×D412yes—
633C2×C4C2×D3×D412yes—
634C2×C4C2×D3×Q812yes—
635C2×C4C2×D3×Q812yes—
636C2×C4D3×#4112yes—
637C2×C4#101014yes—
638C2×C3—42no—
639C2×C3C22×C22⋊Q811yescharacteristic
640C2×C3#348611yescharacteristic
641C2×C3#350011yescharacteristic
642C2×C3#350212yes—
643C2×C3#351812yes—
644C22—22no—
645C22—24no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22C22×D3×D411yescharacteristic
650C22C22×D3×Q811yescharacteristic
651C22C2×D3×#4111yescharacteristic
652C22C2×#101012yes—
653C22#596412yes—
654C4—22no—
655C4—24no—
656C4—22no—
657C4—22no—
658C4—38no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—64no—
665C4—64no—
666C4—122no—
667C4—124no—
668C4—124no—
669C4D3×C22⋊Q814yes—
670C3#3571111yescharacteristic, complemented, Sylow
671C2—42no—
672C2C2×D3×C22⋊Q811yescharacteristic
673C2#8602211yescharacteristic
674C2#8612712yes—
675C2#8624411yescharacteristic
676C2#8625812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2396C2
prime radicalC24×C3C24
hypercenter#2396D3
hypocenterC3#35711
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2396

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic6C2semidirect
4. #82480C2semidirect
6. #78181C2semidirect
8. #77748C2semidirect
10. #77191C2semidirect
12. #73722C2semidirect
13. #73674C2semidirect
14. #73637C2semidirect
15. #73637C2semidirect
108. C22×Dic6Q8semidirect
670. C3#35711semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic62
C2×#55162
C3×#23962
#824802
#822642
#781812
#780492
#777482
#772882
#771912
#767782
#737222
#736742
#736372
#734922
C22⋊C4×Dic62
#357113