Finite Group Databasebeta

← #13390167

Subgroups — #13390167

Order 1792.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic14C211yesmaximal, characteristic, complemented
2C2×C4×C7×D8C211yesmaximal, characteristic
3#11739821C211yesmaximal, characteristic, complemented
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729791C212yesmaximal
6#11729358C211yesmaximal, characteristic, complemented
7#11728371C212yesmaximal, complemented
8#11727921C212yesmaximal, complemented
9#11727917C212yesmaximal, complemented
10#11727542C211yesmaximal, characteristic, complemented
11#11727531C211yesmaximal, characteristic, complemented
12C23×Dic14—21no—
13C22×C7×D8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×D4×Dic7C2212yes—
17C2×#6487—21no—
18C2×#6487—21no—
19D4×Dic14—24no—
20D4×Dic14—24no—
21C2×#6451C2212yes—
22C2×#6441—22no—
23C2×#6432C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C4×C7×D8—24no—
26C2×C7×#107C2211yescharacteristic
27C2×C7×#102C2211yescharacteristic
28C2×C7×#102C2211yescharacteristic
29C2×C4×C7×C8C2211yescharacteristic
30#89993C2212yes—
31C2×#6406C2212yes—
32C2×#6406C2212yes—
33#89984—21no—
34#89779—24no—
35#89745C2212yes—
36C2×#6381C2212yes—
37#89720—22no—
38#89707C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40#89346C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43#89345C2212yes—
44#89324C2212yes—
45#24326—71nomaximal, Sylow
46C22×C7×D4C2311yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C23×Dic7—22no—
51C22×Dic14—28no—
52C22×Dic14—21no—
53C22×Dic14—22no—
54C22×Dic14—44no—
55C22×Dic14—44no—
56C22×Dic14D411yescharacteristic
57C2×C7×D8—22no—
58C2×C7×D8—22no—
59C2×C7×D8Q814yescomplemented
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C4×C7×D4D414yes—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—22no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—22no—
74D4×Dic7—24no—
75D4×Dic7—24no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7C2312yes—
81C2×C4⋊Dic7C2312yes—
82#6487—28no—
83#6487—44no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—22no—
86C2×Dic7⋊C4D412yes—
87C2×C4×Dic7C2312yes—
88C22×C7⋊C8C2312yes—
89#6451—24no—
90#6451—24no—
91#6441—216no—
92#6432—22no—
93#6432D414yes—
94C4×Dic14—22no—
95C4×Dic14—22no—
96C7×#107—22no—
97C7×#102—22no—
98C7×#102—22no—
99C4×C7×C8—22no—
100#6406—24no—
101#6406—24no—
102C7⋊#95—22no—
103#6381—24no—
104C4⋊(C7⋊C8)—22no—
105C2×D4×Q8—71no—
106C2×#474—71no—
107C2×C4×D8D711yescharacteristic
108#2538—71no—
109#2057—72no—
110#2023—72no—
111#1990—71no—
112#1984—71no—
113#1879—72no—
114#1862—72no—
115#1802—71no—
116C24×C7—21no—
117C24×C7—21no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4C2×D412yes—
127C2×C7×D4C2×Q812yes—
128C2×C7×D4C2×Q812yescomplemented
129C2×C7×D4#4112yes—
130C22×C4×C7—21no—
131C22×C4×C7—24no—
132C22×C4×C7—21no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—28no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—44no—
145C22×Dic7—44no—
146C22×Dic7C2×D412yes—
147C22×Dic7C2×D412yes—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—28no—
151C2×Dic14—24no—
152C2×Dic14—432no—
153C2×Dic14—44no—
154C2×Dic14—42no—
155C2×Dic14—42no—
156C2×Dic14—44no—
157C2×Dic14D814yescomplemented
158C7×D8—28no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8C2×Q812yescomplemented
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4C2×D412yes—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—42no—
171C42×C7C2×D412yes—
172C42×C7C2×D412yes—
173C22⋊Dic7—28no—
174C22⋊Dic7—44no—
175C22⋊Dic7—82no—
176C4⋊Dic7—22no—
177C4⋊Dic7—24no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—42no—
182C4⋊Dic7—81no—
183C4⋊Dic7C2×D414yes—
184Dic7⋊C4—28no—
185Dic7⋊C4—24no—
186Dic7⋊C4—44no—
187C4×Dic7—22no—
188C4×Dic7—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C23×Q8—141no—
193C22×D8C2×D711yescharacteristic
194D4×Q8—144no—
195D4×Q8—144no—
196C2×C4⋊Q8—72no—
197C2×C4⋊Q8—71no—
198C2×C22⋊Q8—142no—
199C2×C22⋊Q8—141no—
200C2×C22⋊Q8—141no—
201C2×C4×Q8—71no—
202C2×C4×D4—72no—
203C2×C4×D4C2×D711yescharacteristic
204C2×C4×D4C2×D711yescharacteristic
205#474—144no—
206C4×D8—24no—
207C2×#107—72no—
208C2×#107C2×D711yescharacteristic
209C2×C4⋊C8—71no—
210C2×#102—72no—
211C2×#102—72no—
212C2×#102C2×D711yescharacteristic
213C2×#102C2×D711yescharacteristic
214C2×C4×C8C2×D711yescharacteristic
215#396—142no—
216#390—72no—
217#389—72no—
218#384—71no—
219#380—141no—
220#340—72no—
221#340—72no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7C22×D411yescharacteristic
240C7×D4—22no—
241C7×D4—22no—
242C7×D4—22no—
243C7×D4—22no—
244C7×D4—22no—
245C7×D4—41no—
246C7×D4—41no—
247C7×D4C22⋊Q814yescomplemented
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—22no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—48no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—84no—
258C2×C4×C7#12012yes—
259C2×C4×C7#12012yes—
260C2×C4×C7C2×D812yes—
261C2×C4×C7AGL(1,ℤ/8)12yes—
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×Q811yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C72−1+411yescharacteristic
267C2×C4×C72−1+411yescharacteristic
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—24no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—416no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7C2×D814yes—
281Dic14—28no—
282Dic14—416no—
283Dic14—48no—
284C7×C8—22no—
285C7×C8—41no—
286C7⋊C8—42no—
287C22×Q8—71no—
288C22×Q8—142no—
289C22×Q8—148no—
290C22×Q8—141no—
291C22×Q8—284no—
292C22×Q8—284no—
293C22×D4C22×D711yescharacteristic
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—142no—
298C2×D8—22no—
299C2×D8—22no—
300C2×D8Dic1414yescomplemented
301C22×C8—72no—
302C22×C8C22×D711yescharacteristic
303C22×C8C22×D711yescharacteristic
304C4⋊Q8—74no—
305C4⋊Q8—144no—
306C4⋊Q8—144no—
307C4⋊Q8—142no—
308C22⋊Q8—1416no—
309C22⋊Q8—148no—
310C22⋊Q8—284no—
311C4×Q8—142no—
312C4×Q8—142no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—144no—
317C4×D4—144no—
318C4×D4C22⋊D714yes—
319C2×C4⋊C4—72no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—72no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—142no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—142no—
334C2×C22⋊C4—142no—
335C2×C42—72no—
336C2×C42C22×D711yescharacteristic
337#107—22no—
338#107—144no—
339C4⋊C8—142no—
340#102—22no—
341#102—22no—
342#102—144no—
343#102—144no—
344C4×C8—22no—
345#95—142no—
346C22×C7—24no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—81no—
360C22×C7C2×#12011yescharacteristic
361C22×C7#54811yescharacteristic
362C22×C7D4×Q811yescharacteristic
363C22×C7C22×D811yescharacteristic
364C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C7#57711yescharacteristic
366C22×C7#57811yescharacteristic
367C4×C7—22no—
368C4×C7—42no—
369C4×C7—81no—
370C4×C7#44714yes—
371C4×C7C2×C22⋊Q811yescharacteristic
372C4×C7#53611yescharacteristic
373C4×C7#54811yescharacteristic
374C4×C7D4×Q811yescharacteristic
375Dic7—24no—
376Dic7—44no—
377Dic7—42no—
378Dic7—82no—
379C24—21no—
380C24—21no—
381C2×Q8—74no—
382C2×Q8—148no—
383C2×Q8—142no—
384C2×Q8—142no—
385C2×Q8—144no—
386C2×Q8—284no—
387C2×Q8—284no—
388C2×Q8—2832no—
389C2×Q8—282no—
390C2×Q8—282no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4C2×Dic1412yes—
400C2×D4C2×Dic1412yescomplemented
401C2×D4#115712yes—
402C2×D4C2×C22⋊D712yes—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—72no—
412C22×C4—72no—
413C22×C4—148no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—284no—
417C22×C4—284no—
418C22×C4D4×D711yescharacteristic
419C22×C4D4×D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421D8—28no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—142no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8C2×Dic1412yescomplemented
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—74no—
435C4⋊C4—148no—
436C4⋊C4—144no—
437C4⋊C4—142no—
438C4⋊C4—144no—
439C4⋊C4—142no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—284no—
443C4⋊C4—282no—
444C4⋊C4—561no—
445C4⋊C4C2×C22⋊D712yes—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—148no—
449C22⋊C4—284no—
450C22⋊C4—562no—
451C42—142no—
452C42—142no—
453C42D4×D712yes—
454C42C2×C22⋊D712yes—
455C2×C7—44no—
456C2×C7—82no—
457C2×C7#240911yescharacteristic
458C2×C7C2×#44711yescharacteristic
459C2×C7#301311yescharacteristic
460C2×C7#335811yescharacteristic
461C2×C7#336711yescharacteristic
462C2×C7Q8×D811yescharacteristic
463C2×C7#338411yescharacteristic
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C2×D4×D711yescharacteristic
479Q8—148no—
480Q8—2816no—
481Q8—288no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4#648714yescomplemented
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—48no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—74no—
500C2×C4—84no—
501C2×C4—142no—
502C2×C4—142no—
503C2×C4—142no—
504C2×C4—142no—
505C2×C4—142no—
506C2×C4—144no—
507C2×C4—2816no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4D7×D812yes—
514C2×C4#647412yes—
515C2×C4#650012yes—
516C2×C4#650012yes—
517C2×C4C22×Dic1411yescharacteristic
518C2×C4C2×#115711yescharacteristic
519C2×C4C2×D4×D711yescharacteristic
520C2×C4#655411yescharacteristic
521C2×C4#655411yescharacteristic
522C2×C4C22×C22⋊D711yescharacteristic
523C8—22no—
524C8—41no—
525C8—282no—
526C7#2432611yescharacteristic, complemented, Sylow
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22D4×Dic1411yescharacteristic
542C22#9024311yescharacteristic
543C22C2×D7×D811yescharacteristic
544C22C2×#647411yescharacteristic
545C22#9046111yescharacteristic
546C22#9046211yescharacteristic
547C22C2×#650011yescharacteristic
548C4—22no—
549C4—42no—
550C4—81no—
551C4—144no—
552C4—284no—
553C4—282no—
554C4—562no—
555C4#8992814yes—
556C4D4×Dic1411yescharacteristic
557C4#9024311yescharacteristic
558C4C2×#648711yescharacteristic
559C4#9051611yescharacteristic
560C2—44no—
561C2—82no—
562C2#1173701211yescharacteristic
563C2#1173701611yescharacteristic
564C2D8×Dic1411yescharacteristic
565C2#1173702111yescharacteristic
566C2#1173979411yescharacteristic
567C2C2×#8992811yescharacteristic
568C2#1174135811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×D8C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
hypercenterC2×C4×D8D7
hypocenterC7#24326
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic14C2semidirect
3. #11739821C2semidirect
4. C2×#89779C2semidirect
6. #11729358C2semidirect
7. #11728371C2semidirect
8. #11727921C2semidirect
9. #11727917C2semidirect
10. #11727542C2semidirect
11. #11727531C2semidirect
59. C2×C7×D8Q8semidirect
128. C2×C7×D4C2×Q8semidirect
157. C2×Dic14D8semidirect
164. C2×C7×C8C2×Q8semidirect
247. C7×D4C22⋊Q8semidirect
300. C2×D8Dic14semidirect
400. C2×D4C2×Dic14semidirect
430. C2×C8C2×Dic14semidirect
489. D4#6487semidirect
526. C7#24326semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic142
C2×C4×C7×D82
#117398212
C2×#897792
#117297912
#117293582
#117283712
#117279212
#117279172
#117275422
#117275312
#243267