Finite Group Databasebeta

← #13411318

Subgroups — #13411318

Order 1792.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#89997C211yesmaximal, characteristic
3C2×#89794C211yesmaximal, characteristic, complemented
4#11730939C211yesmaximal, characteristic
5#11729886C211yesmaximal, characteristic
6#11729383C211yesmaximal, characteristic, complemented
7#11729382C211yesmaximal, characteristic, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727438C211yesmaximal, characteristic, complemented
10#11727436C211yesmaximal, characteristic, complemented
11#11727436C211yesmaximal, characteristic, complemented
12#11727423C211yesmaximal, characteristic, complemented
13#11727267C211yesmaximal, characteristic
14#11727260C211yesmaximal, characteristic
15#11727204C211yesmaximal, characteristic
16C23×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C7×#391—21no—
23C7×#391—21no—
24C7×#391C2211yescharacteristic
25C7×#391C2211yescharacteristic
26C7×#380—21no—
27C7×#380C2211yescharacteristic
28C4×C7×C22⋊C4—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30#89997—24no—
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36C2×#6409C2211yescharacteristic
37C2×#6409C2211yescharacteristic
38C2×#6409C2211yescharacteristic
39#89984C2211yescharacteristic
40C2×C22⋊(C7⋊C8)C2211yescharacteristic
41#89794—24no—
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89721—21no—
47#89720—21no—
48#89707C2211yescharacteristic
49#89700C2211yescharacteristic
50#89700C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89426—21no—
54#89419—21no—
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#25981—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×Q8—22no—
64C22×C7×Q8—22no—
65C22×C7×Q8—22no—
66C22×C7×Q8—41no—
67C22×C7×Q8—41no—
68C22×C7×Q8Q814yes—
69C22×C7×Q8C2311yescharacteristic
70C22×C7×Q8C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7C2311yescharacteristic
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
75C4×C7×Q8—41no—
76C4×C7×Q8—41no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C7×C22⋊C4D411yescharacteristic
84C2×C42×C7—21no—
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7—21no—
90#6489—22no—
91#6489—41no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×C4×Dic7—21no—
103C2×#1136—22no—
104C2×#1136—22no—
105C2×#1136—41no—
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110#6433—22no—
111#6433—41no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95—21no—
116C7×#95—21no—
117C7×#95D411yescharacteristic
118C7×#95D411yescharacteristic
119#6409—22no—
120#6409—22no—
121#6409—22no—
122#6409—22no—
123#6409—22no—
124#6409—41no—
125#6409—41no—
126#6409D414yes—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95—21no—
130C22⋊(C7⋊C8)—22no—
131#6382—22no—
132#6382—41no—
133#6381—22no—
134#6381—41no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)—41no—
137C2×#479—71no—
138C2×#419—71no—
139#2441—71no—
140Q8×C22⋊C4D711yescharacteristic
141#2095—71no—
142#2066—71no—
143#2066—71no—
144#2026—71no—
145#1865—71no—
146#1863—71no—
147#1854—71no—
148#1851—71no—
149#1809—71no—
150#1803—71no—
151#1787—71no—
152C24×C7C2×D411yescharacteristic
153C2×C7×Q8—24no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—28no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—42no—
164C2×C7×Q8—42no—
165C2×C7×Q8—44no—
166C2×C7×Q8—44no—
167C2×C7×Q8—44no—
168C2×C7×Q8C2×D412yes—
169C2×C7×Q8C2×Q812yes—
170C2×C7×Q8C2×Q812yes—
171C2×C7×Q8#4112yes—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—24no—
178C22×C4×C7—21no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—42no—
182C22×C4×C7—42no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×Q812yes—
189C22×C4×C7#4112yes—
190C22×C4×C7C2411yescharacteristic
191C22×Dic7—21no—
192C22×Dic7—21no—
193C22×Dic7—21no—
194C22×Dic7—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—42no—
200C7×C4⋊C4—42no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—41no—
203C7×C22⋊C4—42no—
204C42×C7—21no—
205C42×C7—21no—
206C42×C7—21no—
207C42×C7—21no—
208C42×C7—42no—
209C42×C7—42no—
210C22⋊Dic7—42no—
211C22⋊Dic7—81no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—42no—
220C4⋊Dic7—81no—
221C4⋊Dic7C2×D412yes—
222Dic7⋊C4—42no—
223Dic7⋊C4—42no—
224Dic7⋊C4—42no—
225Dic7⋊C4—81no—
226C4×Dic7—42no—
227#1136—44no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8C2×D412yes—
240C23×Q8C2×D711yescharacteristic
241C22×M16—71no—
242C2×#125—71no—
243C2×C4×Q8C2×D711yescharacteristic
244C2×C4×Q8C2×D711yescharacteristic
245C2×#117—71no—
246#479—144no—
247C2×#107—71no—
248C2×#107—71no—
249C2×#106—71no—
250C2×#106—71no—
251C2×C4⋊C8—71no—
252C2×C4⋊C8—71no—
253#419—144no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×C22⋊C8—71no—
263#399—141no—
264#396—141no—
265#391—21no—
266#391—21no—
267#391C2×D711yescharacteristic
268#391C2×D711yescharacteristic
269#389—71no—
270#389—71no—
271#389—141no—
272#387—141no—
273#384—71no—
274#380—21no—
275#380—71no—
276#380C2×D711yescharacteristic
277C4×C22⋊C4—21no—
278C4×C22⋊C4C2×D711yescharacteristic
279#340—71no—
280#340—71no—
281#340—71no—
282#340—71no—
283#336—71no—
284#336—71no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7C22⋊Q812yes—
292C23×C7C22×D411yescharacteristic
293C7×Q8—22no—
294C7×Q8—22no—
295C7×Q8—22no—
296C7×Q8—22no—
297C7×Q8—22no—
298C7×Q8—48no—
299C7×Q8—41no—
300C7×Q8—41no—
301C7×Q8C22⋊Q814yes—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—22no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—21no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—21no—
322C2×C4×C7—21no—
323C2×C4×C7—48no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7C22⋊Q812yes—
333C2×C4×C7#13712yes—
334C2×C4×C7C22×D411yescharacteristic
335C2×C4×C7C22×Q811yescharacteristic
336C2×C4×C7C2×#4111yescharacteristic
337C2×C4×C7C2×#4111yescharacteristic
338C2×C4×C7H(5,𝔽2)11yescharacteristic
339C2×C4×C7H(5,𝔽2)11yescharacteristic
340C2×Dic7—22no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C7⋊C8—22no—
353C7⋊C8—41no—
354C7⋊C8—41no—
355C7⋊C8—41no—
356C22×Q8—22no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—41no—
361C22×Q8—41no—
362C22×Q8Dic1414yes—
363C22×Q8C22×D711yescharacteristic
364C22×Q8C22×D711yescharacteristic
365C23×C4—21no—
366C23×C4C22×D711yescharacteristic
367C2×M16—142no—
368C2×M16—142no—
369C2×M16—281no—
370C22×C8—71no—
371C22×C8—71no—
372C22×C8—71no—
373C22×C8—71no—
374#125—142no—
375#125—281no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8—41no—
379C4×Q8—41no—
380#117—142no—
381#117—281no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4C22⋊D711yescharacteristic
400C2×C22⋊C4C22⋊D711yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—141no—
404C2×C42C22×D711yescharacteristic
405C2×C42C22×D711yescharacteristic
406#107—142no—
407#107—281no—
408#106—142no—
409#106—281no—
410C4⋊C8—142no—
411C4⋊C8—281no—
412#103—74no—
413#103—142no—
414#103—142no—
415#103—142no—
416#103—142no—
417#103—142no—
418#103—281no—
419#103—281no—
420C22⋊C8—142no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95C22⋊D711yescharacteristic
430#95C22⋊D711yescharacteristic
431C22×C7—22no—
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—41no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7C2×C22⋊Q811yescharacteristic
439C22×C7D4211yescharacteristic
440C22×C7#54611yescharacteristic
441C22×C7C2×#13711yescharacteristic
442C22×C7#57511yescharacteristic
443C22×C7#57711yescharacteristic
444C22×C7#57811yescharacteristic
445C4×C7—22no—
446C4×C7—24no—
447C4×C7—22no—
448C4×C7—42no—
449C4×C7—41no—
450C4×C7—41no—
451C4×C7—41no—
452C4×C7—81no—
453C4×C7—81no—
454C4×C7C2×C22⋊Q811yescharacteristic
455C4×C7#54311yescharacteristic
456C4×C7#54611yescharacteristic
457C4×C7#55111yescharacteristic
458Dic7—42no—
459Dic7—81no—
460Dic7—81no—
461Dic7—81no—
462C24C2×C22⋊D711yescharacteristic
463C2×Q8—24no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—28no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—44no—
478C2×Q8C2×Dic1412yes—
479C2×Q8C2×Dic1412yes—
480C2×Q8D4×D712yes—
481C2×Q8#115912yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4—141no—
497C22×C4C2×Dic1412yes—
498C22×C4#115712yes—
499C22×C4C2×C22⋊D711yescharacteristic
500C22×C4C2×C22⋊D711yescharacteristic
501C22×C4C2×C22⋊D711yescharacteristic
502C22×C4C2×C22⋊D711yescharacteristic
503C22×C4C2×C22⋊D711yescharacteristic
504C22×C4C23×D711yescharacteristic
505M16—284no—
506C2×C8—72no—
507C2×C8—141no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8—141no—
513C2×C8—141no—
514C2×C8—141no—
515C2×C8—141no—
516C2×C8—141no—
517C2×C8—141no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—72no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C4⋊C4—141no—
529C4⋊C4—141no—
530C4⋊C4—141no—
531C4⋊C4—141no—
532C4⋊C4—282no—
533C4⋊C4—282no—
534C4⋊C4—282no—
535C4⋊C4—282no—
536C4⋊C4—561no—
537C4⋊C4—561no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—282no—
542C22⋊C4—561no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—42no—
549C42—282no—
550C2×C7—42no—
551C2×C7#243711yescharacteristic
552C2×C7#307811yescharacteristic
553C2×C7#308211yescharacteristic
554C2×C7#327111yescharacteristic
555C2×C7#327611yescharacteristic
556C2×C7#332011yescharacteristic
557C2×C7#332711yescharacteristic
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#648712yes—
565C23C22×C22⋊D711yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—48no—
572Q8—41no—
573Q8—41no—
574Q8#644114yes—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—142no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4#648712yes—
618C2×C4#650512yes—
619C2×C4C22×Dic1411yescharacteristic
620C2×C4C2×#115711yescharacteristic
621C2×C4C2×D4×D711yescharacteristic
622C2×C4C2×#115911yescharacteristic
623C2×C4#654811yescharacteristic
624C2×C4#654811yescharacteristic
625C8—142no—
626C8—281no—
627C8—281no—
628C8—281no—
629C7#2598111yescharacteristic, complemented, Sylow
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#648711yescharacteristic
638C22D4×C22⋊D711yescharacteristic
639C22#9049711yescharacteristic
640C22C2×#650511yescharacteristic
641C22#9051311yescharacteristic
642C22#9052811yescharacteristic
643C22#9052911yescharacteristic
644C4—22no—
645C4—22no—
646C4—24no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—282no—
654C4—561no—
655C4—561no—
656C4—561no—
657C4C2×#644111yescharacteristic
658C4#9017411yescharacteristic
659C4#9018111yescharacteristic
660C4#9018611yescharacteristic
661C2—42no—
662C2#1173955511yescharacteristic
663C2#1173987611yescharacteristic
664C2#1173999511yescharacteristic
665C2#1174098611yescharacteristic
666C2#1174099211yescharacteristic
667C2#1174099811yescharacteristic
668C2#1174100511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×Q8×C22⋊C4C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterQ8×C22⋊C4D7
hypocenterC7#25981
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#89794C2semidirect
6. #11729383C2semidirect
7. #11729382C2semidirect
8. #11729380C2semidirect
9. #11727438C2semidirect
10. #11727436C2semidirect
11. #11727436C2semidirect
12. #11727423C2semidirect
629. C7#25981semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C22⋊C42
C2×#899972
C2×#897942
#117309392
#117298862
#117293832
#117293822
#117293802
#117274382
#117274362
#117274232
#117272672
#117272602
#117272042
#259817