Finite Group Databasebeta

← #31934092

Subgroups — #31934092

Order 18176.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985922C214yesmaximal, complemented
2C2×#17985902C211yesmaximal, characteristic
3C2×#17985902C211yesmaximal, characteristic
4C2×#17985895C211yesmaximal, characteristic, complemented
5C2×#17985869C211yesmaximal, characteristic, complemented
6C2×C4×Dic71⋊C4C211yesmaximal, characteristic, complemented
7#22586585C214yesmaximal, complemented
8#22586569C211yesmaximal, characteristic, complemented
9C2×#390×C71C211yesmaximal, characteristic
10C22×C4⋊Dic71C2211yescharacteristic
11C22×C4⋊Dic71C2214yes—
12C22×C4⋊Dic71C2211yescharacteristic
13C22×Dic71⋊C4C2212yes—
14C22×Dic71⋊C4C2212yes—
15C22×C22⋊Dic71C2212yes—
16C22×C22⋊Dic71C2214yes—
17C22×C22⋊Dic71C2212yes—
18C22×C4×Dic71C2211yescharacteristic
19C22×C4×Dic71C2211yescharacteristic
20#17985922—216no—
21#17985902—24no—
22#17985902—24no—
23#17985895—24no—
24#17985869—24no—
25C4×Dic71⋊C4—24no—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2214yes—
28C2×C71⋊#95C2212yes—
29C23×D4×C71C2211yescharacteristic
30C22×C4⋊C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71C2214yes—
32C22×C42×C71C2211yescharacteristic
33#390×C71—24no—
34C23×Dic71C2312yes—
35C23×Dic71C2311yescharacteristic
36C23×Dic71C2312yes—
37C23×Dic71C2312yes—
38C23×Dic71C2311yescharacteristic
39C2×C4⋊Dic71—28no—
40C2×C4⋊Dic71—22no—
41C2×C4⋊Dic71—22no—
42C2×C4⋊Dic71—22no—
43C2×C4⋊Dic71—28no—
44C2×C4⋊Dic71—22no—
45C2×C4⋊Dic71—22no—
46C2×C4⋊Dic71—44no—
47C2×C4⋊Dic71D414yes—
48C2×Dic71⋊C4—24no—
49C2×Dic71⋊C4—28no—
50C2×Dic71⋊C4—24no—
51C2×Dic71⋊C4—24no—
52C2×Dic71⋊C4D418yescomplemented
53C2×C22⋊Dic71—28no—
54C2×C22⋊Dic71—24no—
55C2×C22⋊Dic71—24no—
56C2×C22⋊Dic71—28no—
57C2×C22⋊Dic71—28no—
58C2×C22⋊Dic71—44no—
59C2×C22⋊Dic71—44no—
60C2×C4×Dic71—22no—
61C2×C4×Dic71—22no—
62C2×C4×Dic71—22no—
63C2×C4×Dic71—22no—
64C2×C4×Dic71—22no—
65C2×C4×Dic71D414yes—
66C71⋊#95—24no—
67C71⋊#95—28no—
68C71⋊#95—24no—
69C25×C71C2312yes—
70C22×D4×C71—22no—
71C22×D4×C71—22no—
72C22×D4×C71—22no—
73C22×D4×C71—22no—
74C22×D4×C71—22no—
75C22×D4×C71—22no—
76C22×D4×C71Q814yes—
77C23×C4×C71C2312yes—
78C23×C4×C71C2312yes—
79C23×C4×C71C2311yescharacteristic
80C2×C4⋊C4×C71—22no—
81C2×C4⋊C4×C71—22no—
82C2×C4⋊C4×C71—22no—
83C2×C22⋊C4×C71—28no—
84C2×C22⋊C4×C71—28no—
85C2×C22⋊C4×C71—28no—
86C2×C42×C71—22no—
87C2×C42×C71—22no—
88C2×C42×C71D414yes—
89C22×Dic71—21no—
90C22×Dic71—21no—
91C22×Dic71—22no—
92C22×Dic71—21no—
93C22×Dic71—22no—
94C22×Dic71—22no—
95C22×Dic71—22no—
96C22×Dic71—24no—
97C22×Dic71—21no—
98C22×Dic71—22no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—24no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—24no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—24no—
111C22×Dic71C2×D412yes—
112C22×Dic71C2×D414yes—
113C22×Dic71C2×D414yes—
114C22×Dic71C2×D412yes—
115C22×Dic71C2×D414yes—
116C4⋊Dic71—28no—
117C4⋊Dic71—416no—
118C4⋊Dic71—44no—
119Dic71⋊C4—216no—
120Dic71⋊C4—48no—
121C22⋊Dic71—416no—
122C22⋊Dic71—48no—
123C22⋊Dic71—48no—
124C4×Dic71—28no—
125C4×Dic71—44no—
126C24×C71—22no—
127C24×C71—24no—
128C24×C71—22no—
129C24×C71—22no—
130C24×C71—24no—
131C24×C71—22no—
132C24×C71—22no—
133C24×C71—22no—
134C24×C71—24no—
135C24×C71—24no—
136C24×C71C2×Q814yes—
137C24×C71C2411yescharacteristic
138C2×D4×C71—28no—
139C2×D4×C71—28no—
140C2×D4×C71—28no—
141C2×D4×C71—44no—
142C2×D4×C71—44no—
143C2×D4×C71—44no—
144C2×D4×C71—44no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—24no—
151C22×C4×C71—22no—
152C22×C4×C71—24no—
153C22×C4×C71—22no—
154C22×C4×C71C2×D414yes—
155C22×C4×C71C2×D412yes—
156C22×C4×C71C2×D412yes—
157C22×C4×C71C2×D414yes—
158C22×C4×C71C2×D412yes—
159C22×C4×C71C2×Q812yes—
160C22×C4×C71#4114yes—
161C22×C4×C71#4112yes—
162C22×C4×C71#4112yes—
163C22×C4×C71#4112yes—
164C4⋊C4×C71—44no—
165C22⋊C4×C71—416no—
166C42×C71—28no—
167C23×C71—24no—
168C23×C71—24no—
169C23×C71—24no—
170C23×C71—28no—
171C23×C71—28no—
172C23×C71—44no—
173C23×C71—42no—
174C23×C71—42no—
175C23×C71—42no—
176C23×C71—42no—
177C23×C71—42no—
178C23×C71—42no—
179C23×C71—44no—
180C23×C71—42no—
181C23×C71—42no—
182C23×C71—44no—
183C23×C71—44no—
184C23×C71—44no—
185C23×C71—44no—
186C23×C71—44no—
187C23×C71—44no—
188C23×C71—44no—
189C23×C71—44no—
190C23×C71C22×D411yescharacteristic
191C23×C71C22×D411yescharacteristic
192C23×C71C22×D412yes—
193C23×C71C22×D411yescharacteristic
194C23×C71C22×Q811yescharacteristic
195C23×C71C2×#4111yescharacteristic
196C23×C71C2×#4111yescharacteristic
197C23×C71C2×#4112yes—
198C23×C71C2×#4111yescharacteristic
199C23×C71H(5,𝔽2)12yes—
200C23×C712−1+412yes—
201D4×C71—416no—
202C2×C4×C71—24no—
203C2×C4×C71—24no—
204C2×C4×C71—24no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—44no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—44no—
213C2×C4×C71#12118yes—
214C2×C4×C71#12114yes—
215C2×C4×C71#12114yes—
216C2×C4×C71#12114yes—
217C2×C4×C71C22⋊Q814yes—
218C2×C4×C71C22⋊Q814yes—
219C2×C4×C71C22⋊Q814yes—
220C2×C4×C71#12614yes—
221C2×Dic71—22no—
222C2×Dic71—24no—
223C2×Dic71—22no—
224C2×Dic71—24no—
225C2×Dic71—22no—
226C2×Dic71—24no—
227C2×Dic71—24no—
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—24no—
231C2×Dic71—28no—
232C2×Dic71—24no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—44no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—44no—
246C2×Dic71—44no—
247C2×Dic71#12714yes—
248C22×C71—44no—
249C22×C71—44no—
250C22×C71—48no—
251C22×C71—48no—
252C22×C71—44no—
253C22×C71—82no—
254C22×C71—82no—
255C22×C71—84no—
256C22×C71—84no—
257C22×C71—82no—
258C22×C71—82no—
259C22×C71C2×#12111yescharacteristic
260C22×C71C2×#12112yes—
261C22×C71C2×#12111yescharacteristic
262C22×C71C2×#12111yescharacteristic
263C22×C71C2×C22⋊Q811yescharacteristic
264C22×C71C2×C22⋊Q811yescharacteristic
265C22×C71C2×C22⋊Q811yescharacteristic
266C22×C71C2×#12611yescharacteristic
267C22×C71C2×#12711yescharacteristic
268C22×C71#53211yescharacteristic
269C22×C71#53212yes—
270C22×C71#53511yescharacteristic
271C22×C71#53611yescharacteristic
272C22×C71#53712yes—
273C22×C71#54212yes—
274C22×C71D4212yes—
275C22×C71#54712yes—
276C22×C71#54812yes—
277C22×C71D4×Q812yes—
278C22×C71#55012yes—
279C22×C71#55312yes—
280C22×C71#55412yes—
281C22×C71#55612yes—
282C4×C71—28no—
283C4×C71—82no—
284C4×C71#39618yescomplemented
285Dic71—24no—
286Dic71—42no—
287Dic71—44no—
288Dic71—44no—
289Dic71—82no—
290#15134—711nomaximal, Sylow
291C2×C71—84no—
292C2×C71C2×#39611yescharacteristic
293C2×C71#241612yes—
294C2×C71#245212yes—
295C2×C71#251212yes—
296C2×C71#253812yes—
297C2×C71#254111yescharacteristic
298C2×C71#254211yescharacteristic
299C2×C71#254511yescharacteristic
300C2×C71#256611yescharacteristic
301C2×C71#262011yescharacteristic
302C2×C71#285611yescharacteristic
303C2×#396—714no—
304C2×#390—711no—
305C2×#390—711no—
306C2×#390D7111yescharacteristic
307C2×#389—711no—
308C2×#384—711no—
309C2×C4×C4⋊C4—711no—
310#1439—714no—
311C4⋊#95—711no—
312C71#1513411yescharacteristic, complemented, Sylow
313C23×D4C2×D7111yescharacteristic
314C22×C4⋊C4—714no—
315C22×C4⋊C4—712no—
316C22×C4⋊C4—711no—
317C22×C4⋊C4—712no—
318C22×C4⋊C4—711no—
319C22×C4⋊C4C2×D7111yescharacteristic
320C22×C22⋊C4—712no—
321C22×C22⋊C4—714no—
322C22×C22⋊C4—712no—
323C22×C22⋊C4C2×D7114yes—
324C22×C42—711no—
325C22×C42—711no—
326C22×C42C2×D7111yescharacteristic
327#396—14216no—
328#390—24no—
329#390—1424no—
330#390—1424no—
331#389—1424no—
332#384—1424no—
333C4×C4⋊C4—1424no—
334C2×#95—714no—
335C2×#95—712no—
336C2×#95—712no—
337C25C22×D7112yes—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4Dic14214yes—
345C23×C4—711no—
346C23×C4—712no—
347C23×C4—712no—
348C23×C4—711no—
349C23×C4—712no—
350C23×C4C22×D7112yes—
351C23×C4C22×D7112yes—
352C23×C4C22×D7111yescharacteristic
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—2844no—
357C2×C4⋊C4—714no—
358C2×C4⋊C4—718no—
359C2×C4⋊C4—1428no—
360C2×C4⋊C4—1424no—
361C2×C4⋊C4—1428no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4—1422no—
364C2×C4⋊C4—1428no—
365C2×C4⋊C4—1422no—
366C2×C4⋊C4—1422no—
367C2×C4⋊C4—1424no—
368C2×C4⋊C4—1424no—
369C2×C4⋊C4—1422no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—2844no—
374C2×C22⋊C4—2844no—
375C2×C22⋊C4—1424no—
376C2×C22⋊C4—1424no—
377C2×C22⋊C4—1428no—
378C2×C22⋊C4—1428no—
379C2×C22⋊C4—1428no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—714no—
383C2×C42—1422no—
384C2×C42—1422no—
385C2×C42—1422no—
386C2×C42—1422no—
387C2×C42—1422no—
388C2×C42C22⋊D7114yes—
389#95—1424no—
390#95—1428no—
391#95—1424no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—24no—
401C24—24no—
402C24C2×Dic14214yes—
403C24C23×D7111yescharacteristic
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—712no—
421C22×C4—714no—
422C22×C4—712no—
423C22×C4—714no—
424C22×C4—714no—
425C22×C4—1424no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1424no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1424no—
433C22×C4—1421no—
434C22×C4—1422no—
435C22×C4—1421no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1422no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1424no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4—1422no—
447C22×C4D4×D7114yes—
448C22×C4D4×D7112yes—
449C22×C4D4×D7112yes—
450C22×C4C2×C22⋊D7114yes—
451C22×C4C2×C22⋊D7112yes—
452C22×C4C2×Dic14212yes—
453C22×C4#1176586414yes—
454C22×C4#1176586412yes—
455C22×C4#1176586412yes—
456C22×C4#1176586512yes—
457C4⋊C4—44no—
458C4⋊C4—28416no—
459C4⋊C4—2848no—
460C4⋊C4—2844no—
461C4⋊C4—14216no—
462C4⋊C4—1428no—
463C22⋊C4—416no—
464C22⋊C4—2848no—
465C22⋊C4—2848no—
466C22⋊C4—28416no—
467C42—28no—
468C42—2844no—
469C42—1428no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23C2×D4×D7111yescharacteristic
494C23C2×D4×D7111yescharacteristic
495C23C2×D4×D7112yes—
496C23C22×C22⋊D7111yescharacteristic
497C23C22×Dic14211yescharacteristic
498C23C2×#1176586411yescharacteristic
499C23C2×#1176586411yescharacteristic
500C23C2×#1176586412yes—
501C23C2×#1176586511yescharacteristic
502C23#1455617412yes—
503C23#1455617512yes—
504D4—416no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—2842no—
517C2×C4—2842no—
518C2×C4—2842no—
519C2×C4—2842no—
520C2×C4—2842no—
521C2×C4—2844no—
522C2×C4—2842no—
523C2×C4—2842no—
524C2×C4—2842no—
525C2×C4—2842no—
526C2×C4—2844no—
527C2×C4—2844no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—714no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1424no—
534C2×C4—1428no—
535C2×C4—1424no—
536C2×C4—1422no—
537C2×C4—1424no—
538C2×C4—1424no—
539C2×C4—1422no—
540C2×C4—1424no—
541C2×C4—1422no—
542C2×C4—1424no—
543C2×C4#1455609918yes—
544C2×C4#1455609914yes—
545C2×C4#1455610014yes—
546C2×C4#1455610114yes—
547C2×C4#1455610514yes—
548C2×C4#1455610514yes—
549C2×C4#1455610714yes—
550C2×C4#1455612114yes—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—48no—
555C22—48no—
556C22—82no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—84no—
562C22C2×#1455609912yes—
563C22C2×#1455609911yescharacteristic
564C22C2×#1455610011yescharacteristic
565C22C2×#1455610111yescharacteristic
566C22C2×#1455610511yescharacteristic
567C22C2×#1455610511yescharacteristic
568C22C2×#1455610711yescharacteristic
569C22C2×#1455612111yescharacteristic
570C22#127×D7111yescharacteristic
571C22#1798654711yescharacteristic
572C22#1798654712yes—
573C22#1798656711yescharacteristic
574C22#1798656911yescharacteristic
575C22#1798657712yes—
576C22#1798661812yes—
577C22#1798663312yes—
578C22#1798664912yes—
579C22#1798665512yes—
580C22D4×Dic14212yes—
581C22#1798667012yes—
582C22#1798670112yes—
583C22#1798670612yes—
584C22#1798671912yes—
585C4—28no—
586C4—82no—
587C4—2844no—
588C4—2844no—
589C4—2842no—
590C4—5682no—
591C4—1424no—
592C4#1798592218yescomplemented
593C2—84no—
594C2C2×#1798592211yescharacteristic
595C2#2259178112yes—
596C2#2259219212yes—
597C2#2259292812yes—
598C2#2259326212yes—
599C2#2259328511yescharacteristic
600C2#2259329111yescharacteristic
601C2#2259332211yescharacteristic
602C2#2259354211yescharacteristic
603C2#2259409811yescharacteristic
604C2#2259668211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutator, socle, cosocleC24×C71C24
FittingC2×#390×C71C2
prime radicalC23×D4×C71C22
hypercenterC2×#390D71
hypocenterC71#15134
LCSG ▷ C24×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#17985922C2semidirect
4. C2×#17985895C2semidirect
5. C2×#17985869C2semidirect
6. C2×C4×Dic71⋊C4C2semidirect
7. #22586585C2semidirect
8. #22586569C2semidirect
52. C2×Dic71⋊C4D4semidirect
284. C4×C71#396semidirect
312. C71#15134semidirect
592. C4#17985922semidirect

Maximal subgroups

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

subgroupindex
C2×#179859222
C2×#179859022
C2×#179858952
C2×#179858692
C2×C4×Dic71⋊C42
#225865852
#225865692
C2×#390×C712
#1513471