Finite Group Databasebeta

← #13277263

Subgroups — #13277263

Order 1792.

2054 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C4⋊Q8C211yesmaximal, characteristic
2C22×Q8×Dic7C211yesmaximal, characteristic
3C2×#90002C211yesmaximal, characteristic
4C2×#90002C211yesmaximal, characteristic
5C2×#90002C211yesmaximal, characteristic
6#11729319C214yesmaximal
7#11729318C214yesmaximal
8#11729289C211yesmaximal, characteristic
9C4×C7⋊#95C211yesmaximal, characteristic
10C23×C7×Q8C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C2×C7×C4⋊Q8—24no—
13C2×C7×C4⋊Q8—24no—
14C2×C7×C4⋊Q8C418yescomplemented
15C22×C7×C4⋊C4C2214yes—
16C22×C42×C7C2211yescharacteristic
17C2×Q8×Dic7—24no—
18C2×Q8×Dic7—24no—
19C2×Q8×Dic7—24no—
20C22×C4⋊Dic7C2212yes—
21C22×C4⋊Dic7C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26#90002—24no—
27#90002—24no—
28#90002—24no—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2214yes—
31C2×C7⋊#95C2214yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2214yes—
35C2×C7⋊#95C2212yes—
36#14024—71nomaximal, Sylow
37C22×C7×Q8—22no—
38C22×C7×Q8—22no—
39C22×C7×Q8—22no—
40C22×C7×Q8—22no—
41C22×C7×Q8—22no—
42C22×C7×Q8—22no—
43C22×C7×Q8—22no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8C2×C414yes—
48C22×C7×Q8C2×C414yes—
49C22×C7×Q8D414yes—
50C22×C7×Q8D414yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2312yes—
54C23×C4×C7C2311yescharacteristic
55C23×Dic7C2312yes—
56C23×Dic7C2312yes—
57C23×Dic7C2311yescharacteristic
58C23×Dic7C2312yes—
59C23×Dic7C2311yescharacteristic
60C7×C4⋊Q8—232no—
61C2×C7×C4⋊C4—28no—
62C2×C7×C4⋊C4—28no—
63C2×C7×C4⋊C4C2×C4116yes—
64C2×C42×C7—22no—
65C2×C42×C7—22no—
66C2×C42×C7C2×C414yes—
67Q8×Dic7—232no—
68C2×C4⋊Dic7—24no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—24no—
72C2×C4⋊Dic7—24no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7D414yes—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7D414yes—
86C7⋊#95—28no—
87C7⋊#95—24no—
88C7⋊#95—28no—
89C7⋊#95—28no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95Q818yescomplemented
93C22×C4⋊Q8D711yescharacteristic
94C22×C4×Q8—71no—
95C2×#391—71no—
96C2×#391—71no—
97C2×#391—71no—
98#1441—74no—
99#1436—74no—
100C4⋊#95—71no—
101C4×#95—71no—
102C24×C7C2411yescharacteristic
103C2×C7×Q8—28no—
104C2×C7×Q8—28no—
105C2×C7×Q8—28no—
106C2×C7×Q8—28no—
107C2×C7×Q8—28no—
108C2×C7×Q8—28no—
109C2×C7×Q8—28no—
110C2×C7×Q8—28no—
111C2×C7×Q8—28no—
112C2×C7×Q8—44no—
113C2×C7×Q8—44no—
114C2×C7×Q8—44no—
115C2×C7×Q8—44no—
116C2×C7×Q8C22⋊C4116yes—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7C22×C414yes—
132C22×C4×C7C22×C414yes—
133C22×C4×C7C22×C414yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D414yes—
136C22×C4×C7C2×D414yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×Q814yes—
142C22×C4×C7#4114yes—
143C22×C4×C7#4112yes—
144C22×C4×C7#4112yes—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153C22×Dic7—21no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—24no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—24no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—24no—
167C22×Dic7C2×D414yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×Q814yes—
171C22×Dic7C2×Q814yes—
172C7×C4⋊C4—232no—
173C42×C7—28no—
174C4⋊Dic7—216no—
175C4⋊Dic7—28no—
176C4⋊Dic7—44no—
177C4×Dic7—216no—
178C4×Dic7—28no—
179C4×Dic7—44no—
180C23×Q8C2×D711yescharacteristic
181C23×Q8C2×D711yescharacteristic
182C2×C4⋊Q8—24no—
183C2×C4⋊Q8—24no—
184C2×C4⋊Q8Dic718yescomplemented
185C2×C4×Q8—144no—
186C2×C4×Q8—144no—
187C2×C4×Q8—144no—
188C22×C4⋊C4—71no—
189C22×C4⋊C4—72no—
190C22×C4⋊C4—71no—
191C22×C4⋊C4C2×D714yes—
192C22×C42—71no—
193C22×C42—72no—
194C22×C42—71no—
195C22×C42C2×D711yescharacteristic
196#391—144no—
197#391—144no—
198#391—144no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95—74no—
202C2×#95—72no—
203C2×#95—74no—
204C2×#95—74no—
205C2×#95—72no—
206C23×C7C23×C411yescharacteristic
207C23×C7C22×D411yescharacteristic
208C23×C7C22×D411yescharacteristic
209C23×C7C22×D411yescharacteristic
210C23×C7C22×D411yescharacteristic
211C23×C7C22×Q812yes—
212C23×C7C2×#4111yescharacteristic
213C23×C7C2×#4111yescharacteristic
214C23×C7C2×#4112yes—
215C23×C72−1+412yes—
216C23×C72−1+412yes—
217C7×Q8—232no—
218C7×Q8—416no—
219C2×C4×C7—24no—
220C2×C4×C7—24no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7C2×C22⋊C418yes—
239C2×C4×C7C2×C22⋊C414yes—
240C2×C4×C7C4×D414yes—
241C2×C4×C7C4×D414yes—
242C2×C4×C7C4×Q818yescomplemented
243C2×C4×C7#12014yes—
244C2×C4×C7#12114yes—
245C2×C4×C7#12114yes—
246C2×C4×C7#12314yes—
247C2×Dic7—22no—
248C2×Dic7—22no—
249C2×Dic7—24no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—24no—
255C2×Dic7—22no—
256C2×Dic7—28no—
257C2×Dic7—24no—
258C2×Dic7—24no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—44no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—44no—
272C2×Dic7—44no—
273C2×Dic7C4⋊Q814yescomplemented
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—22no—
283C22×Q8—22no—
284C22×Q8C2×Dic714yes—
285C22×Q8C2×Dic714yes—
286C22×Q8C22⋊D714yes—
287C22×Q8C22⋊D714yes—
288C23×C4—71no—
289C23×C4—71no—
290C23×C4—72no—
291C23×C4—72no—
292C23×C4—72no—
293C23×C4C22×D712yes—
294C23×C4C22×D712yes—
295C23×C4C22×D712yes—
296C23×C4C22×D711yescharacteristic
297C4⋊Q8—232no—
298C4×Q8—1432no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—74no—
302C2×C4⋊C4—144no—
303C2×C4⋊C4—144no—
304C2×C4⋊C4—142no—
305C2×C4⋊C4—144no—
306C2×C4⋊C4—142no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4—142no—
309C2×C4⋊C4—142no—
310C2×C4⋊C4C2×Dic7116yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—74no—
314C2×C42—144no—
315C2×C42—144no—
316C2×C42—142no—
317C2×C42—144no—
318C2×C42—142no—
319C2×C42—142no—
320C2×C42—142no—
321C2×C42—142no—
322C2×C42C2×Dic714yes—
323#95—78no—
324#95—144no—
325#95—144no—
326#95—148no—
327#95—144no—
328#95—148no—
329#95—148no—
330C22×C7C22×C22⋊C411yescharacteristic
331C22×C7C2×C4×D411yescharacteristic
332C22×C7C2×C4×D411yescharacteristic
333C22×C7C2×C4×Q812yes—
334C22×C7#51912yes—
335C22×C7C2×#12011yescharacteristic
336C22×C7C2×#12111yescharacteristic
337C22×C7C2×#12111yescharacteristic
338C22×C7C2×#12311yescharacteristic
339C22×C7C2×C4⋊Q811yescharacteristic
340C22×C7#53211yescharacteristic
341C22×C7#53312yes—
342C22×C7#53611yescharacteristic
343C22×C7#53611yescharacteristic
344C22×C7#54112yes—
345C22×C7#54712yes—
346C22×C7#54812yes—
347C22×C7#54812yes—
348C22×C7D4×Q812yes—
349C22×C7D4×Q812yes—
350C22×C7D4×Q812yes—
351C22×C7#55612yes—
352C22×C7#55712yes—
353C4×C7—28no—
354C4×C7—28no—
355C4×C7—44no—
356C4×C7#38618yes—
357Dic7—24no—
358Dic7—44no—
359Dic7—42no—
360Dic7—44no—
361Dic7—82no—
362C24C23×D711yescharacteristic
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8C22⋊Dic7116yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—72no—
392C22×C4—74no—
393C22×C4—72no—
394C22×C4—74no—
395C22×C4—74no—
396C22×C4—141no—
397C22×C4—142no—
398C22×C4—141no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—141no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—144no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—144no—
408C22×C4—142no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—141no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—144no—
416C22×C4—144no—
417C22×C4—142no—
418C22×C4D4×D712yes—
419C22×C4D4×D712yes—
420C22×C4#115912yes—
421C22×C4#115912yes—
422C22×C4Q8×D714yes—
423C22×C4#116114yes—
424C22×C4C22×Dic714yes—
425C22×C4C22×Dic714yes—
426C22×C4C22×Dic714yes—
427C22×C4C22×Dic712yes—
428C22×C4C2×C22⋊D714yes—
429C22×C4C2×C22⋊D714yes—
430C22×C4C2×C22⋊D712yes—
431C22×C4C2×C22⋊D712yes—
432C4⋊C4—232no—
433C4⋊C4—1416no—
434C4⋊C4—148no—
435C4⋊C4—284no—
436C42—28no—
437C42—1416no—
438C42—148no—
439C42—284no—
440C2×C7C2×#38611yescharacteristic
441C2×C7C4×C4⋊Q811yescharacteristic
442C2×C7#232711yescharacteristic
443C2×C7Q8×C22⋊C412yes—
444C2×C7#236512yes—
445C2×C7#242912yes—
446C2×C7#243211yescharacteristic
447C2×C7#243411yescharacteristic
448C2×C7#244612yes—
449C2×C7#254511yescharacteristic
450C2×C7#254611yescharacteristic
451C23C2×D4×D711yescharacteristic
452C23C2×D4×D711yescharacteristic
453C23C2×#115911yescharacteristic
454C23C2×#115911yescharacteristic
455C23C2×Q8×D712yes—
456C23C2×#116112yes—
457C23#655112yes—
458C23#655112yes—
459C23C23×Dic711yescharacteristic
460C23C22×C22⋊D711yescharacteristic
461C23C22×C22⋊D711yescharacteristic
462Q8—232no—
463Q8—416no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—74no—
484C2×C4—142no—
485C2×C4—144no—
486C2×C4—142no—
487C2×C4—144no—
488C2×C4—142no—
489C2×C4—142no—
490C2×C4—148no—
491C2×C4—144no—
492C2×C4—144no—
493C2×C4—144no—
494C2×C4—142no—
495C2×C4—144no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—284no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—284no—
508C2×C4—282no—
509C2×C4—284no—
510C2×C4D4×Dic714yes—
511C2×C4D4×Dic714yes—
512C2×C4#649814yes—
513C2×C4#650014yes—
514C2×C4#650214yes—
515C2×C4#650214yes—
516C2×C4Q8×Dic718yescomplemented
517C2×C4C2×C22⋊Dic718yes—
518C2×C4C2×C22⋊Dic714yes—
519C7#1402411yescharacteristic, complemented, Sylow
520C22#9041212yes—
521C22D7×C4⋊Q811yescharacteristic
522C22#9041511yescharacteristic
523C22#9041511yescharacteristic
524C22#9041611yescharacteristic
525C22#9041712yes—
526C22#9041812yes—
527C22#9041912yes—
528C22#9042212yes—
529C22#9042312yes—
530C22C2×D4×Dic711yescharacteristic
531C22C2×D4×Dic711yescharacteristic
532C22C2×#649811yescharacteristic
533C22C2×#650011yescharacteristic
534C22C2×#650211yescharacteristic
535C22C2×#650211yescharacteristic
536C22C2×Q8×Dic712yes—
537C22#9050312yes—
538C22Q8×C22⋊D712yes—
539C22Q8×C22⋊D712yes—
540C22#9050812yes—
541C22#9050812yes—
542C22C22×C22⋊Dic711yescharacteristic
543C4—28no—
544C4—28no—
545C4—44no—
546C4—144no—
547C4—284no—
548C4—282no—
549C4—284no—
550C4—562no—
551C4#8999218yes—
552C2Dic7×C4⋊Q811yescharacteristic
553C2#1174067211yescharacteristic
554C2#1174067412yes—
555C2#1174067711yescharacteristic
556C2#1174067912yes—
557C2#1174068211yescharacteristic
558C2#1174068311yescharacteristic
559C2#1174069912yes—
560C2#1174070011yescharacteristic
561C2C2×#8999211yescharacteristic
562C2Q8×C22⋊Dic712yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC22×C7×C4⋊Q8C2
socle, prime radical, cosocleC24×C7C24
hypercenterC22×C4⋊Q8D7
hypocenterC7#14024
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. C2×C7×C4⋊Q8C4semidirect
92. C7⋊#95Q8semidirect
184. C2×C4⋊Q8Dic7semidirect
242. C2×C4×C7C4×Q8semidirect
273. C2×Dic7C4⋊Q8semidirect
516. C2×C4Q8×Dic7semidirect
519. C7#14024semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊Q82
C22×Q8×Dic72
C2×#900022
#117293192
#117293182
#117292892
C4×C7⋊#952
#140247