Finite Group Databasebeta

← #15676737

Subgroups — #15676737

Order 2944.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67290C211yesmaximal, characteristic, complemented
2C2×#10625067C211yesmaximal, characteristic, complemented
3C2×#10625055C211yesmaximal, characteristic, complemented
4C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
5#13092332C211yesmaximal, characteristic, complemented
6#13092326C211yesmaximal, characteristic, complemented
7#13092325C211yesmaximal, characteristic, complemented
8#13092321C211yesmaximal, characteristic, complemented
9#13092317C211yesmaximal, characteristic, complemented
10#13092295C211yesmaximal, characteristic, complemented
11#13092294C211yesmaximal, characteristic, complemented
12#13092289C211yesmaximal, characteristic, complemented
13#13092288C211yesmaximal, characteristic, complemented
14#13092275C211yesmaximal, characteristic, complemented
15C23×#394C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C22×C22⋊D23C2211yescharacteristic
18C22×C22⋊D23C2211yescharacteristic
19C22×C22⋊D23C2211yescharacteristic
20C22×C22⋊D23C2211yescharacteristic
21C23×Dic23C2211yescharacteristic
22C23×Dic23C2211yescharacteristic
23C2×#67290—24no—
24C2×#67290—24no—
25C2×#67290—24no—
26C22×D92C2211yescharacteristic
27C22×C4×D23C2211yescharacteristic
28C22×C4×D23C2211yescharacteristic
29C22×Dic46C2211yescharacteristic
30C2×C4⋊C4×C23C2211yescharacteristic
31C2×C22⋊C4×C23C2211yescharacteristic
32C2×C22⋊C4×C23C2211yescharacteristic
33C2×C22⋊C4×C23C2211yescharacteristic
34C2×C22⋊C4×C23C2211yescharacteristic
35C2×C22⋊Dic23C2211yescharacteristic
36C2×C22⋊Dic23C2211yescharacteristic
37C2×(C2×D23)⋊C4C2211yescharacteristic
38C2×(C2×D23)⋊C4C2211yescharacteristic
39C2×(C2×D23)⋊C4C2211yescharacteristic
40C2×(C2×D23)⋊C4C2211yescharacteristic
41C2×(C2×D23)⋊C4C2211yescharacteristic
42C2×(C2×D23)⋊C4C2211yescharacteristic
43C2×Dic23⋊C4C2211yescharacteristic
44C2×Dic23⋊C4C2211yescharacteristic
45C2×Dic23⋊C4C2211yescharacteristic
46C2×Dic23⋊C4C2211yescharacteristic
47C2×C4×Dic23C2211yescharacteristic
48#10625067—24no—
49#10625055—24no—
50(C22⋊D23)⋊C4—24no—
51C23×#95C2211yescharacteristic
52C23⋊#95C2211yescharacteristic
53C23⋊#95C2211yescharacteristic
54C23⋊#95C2211yescharacteristic
55C23⋊#95C2211yescharacteristic
56C23⋊#95C2211yescharacteristic
57C24×C23C2311yescharacteristic
58C24×C23C2311yescharacteristic
59C23×D23C2311yescharacteristic
60C23×D23C2311yescharacteristic
61C2×D4×C23—22no—
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C22×C4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C2×C22⋊D23—22no—
70C2×C22⋊D23—22no—
71C2×C22⋊D23—22no—
72C2×C22⋊D23—22no—
73C2×C22⋊D23—22no—
74C2×C22⋊D23—22no—
75C2×C22⋊D23—22no—
76C2×C22⋊D23—22no—
77C2×C22⋊D23—22no—
78C2×C22⋊D23—22no—
79C2×C22⋊D23—41no—
80C2×C22⋊D23D414yescomplemented
81C22×Dic23—22no—
82C22×Dic23—22no—
83C22×Dic23—22no—
84C22×Dic23—22no—
85C22×Dic23—22no—
86C22×Dic23D414yescomplemented
87C22×Dic23C2311yescharacteristic
88C22×Dic23C2311yescharacteristic
89C22×Dic23C2311yescharacteristic
90C22×Dic23C2311yescharacteristic
91C22×Dic23C2311yescharacteristic
92C22×Dic23C2311yescharacteristic
93#67290—232no—
94C2×D92—22no—
95C2×D92—22no—
96C2×D92—41no—
97C2×C4×D23—22no—
98C2×C4×D23—22no—
99C2×C4×D23—22no—
100C2×C4×D23—22no—
101C2×C4×D23—22no—
102C2×C4×D23—22no—
103C2×Dic46—22no—
104C2×Dic46—22no—
105C2×Dic46D414yescomplemented
106C4⋊C4×C23—22no—
107C22⋊C4×C23—22no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊Dic23—22no—
112C22⋊Dic23—41no—
113(C2×D23)⋊C4—22no—
114(C2×D23)⋊C4—22no—
115(C2×D23)⋊C4—22no—
116(C2×D23)⋊C4—22no—
117(C2×D23)⋊C4—22no—
118(C2×D23)⋊C4—41no—
119Dic23⋊C4—22no—
120Dic23⋊C4—22no—
121Dic23⋊C4—22no—
122Dic23⋊C4—22no—
123C4×Dic23—22no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23—21no—
129C23×C23—21no—
130C23×C23—21no—
131C23×C23—21no—
132C23×C23—21no—
133C23×C23—21no—
134C23×C23—21no—
135C23×C23—21no—
136C23×C23C2×D412yes—
137C23×C23C2×D412yes—
138C23×C23C2411yescharacteristic
139C22×D23—21no—
140C22×D23—21no—
141C22×D23—21no—
142C22×D23—21no—
143C22×D23—21no—
144C22×D23—21no—
145C22×D23—21no—
146C22×D23—21no—
147C22×D23—21no—
148C22×D23—21no—
149C22×D23—21no—
150C22×D23—21no—
151C22×D23C2×D412yes—
152C22×D23#4112yescomplemented
153D4×C23—28no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23C2×D412yes—
169C22⋊D23—28no—
170C22⋊D23—28no—
171C22⋊D23—28no—
172C22⋊D23—44no—
173C2×Dic23—28no—
174C2×Dic23—21no—
175C2×Dic23—21no—
176C2×Dic23—21no—
177C2×Dic23—21no—
178C2×Dic23—21no—
179C2×Dic23—21no—
180C2×Dic23—28no—
181C2×Dic23—21no—
182C2×Dic23—21no—
183C2×Dic23—21no—
184C2×Dic23—21no—
185C2×Dic23—21no—
186C2×Dic23—21no—
187C2×Dic23C2×D412yes—
188C2×Dic23C2×D412yes—
189C2×Dic23C2×D412yes—
190C2×Dic23C2×D412yes—
191C2×Dic23C2×D412yes—
192C2×Dic23#4112yescomplemented
193D92—44no—
194C4×D23—28no—
195C4×D23—44no—
196Dic46—28no—
197#2407—231nomaximal, Sylow
198C22×C23—22no—
199C22×C23—22no—
200C22×C23—22no—
201C22×C23—22no—
202C22×C23—22no—
203C22×C23—22no—
204C22×C23—41no—
205C22×C23—41no—
206C22×C23—41no—
207C22×C23—41no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—41no—
211C22×C23—41no—
212C22×C23C22×D411yescharacteristic
213C22×C23C22×D411yescharacteristic
214C22×C23C22×D411yescharacteristic
215C22×C23C2×#4111yescharacteristic
216C22×C23C2×#4111yescharacteristic
217C22×C23H(5,𝔽2)11yescharacteristic
218C22×C232−1+411yescharacteristic
219C2×D23—22no—
220C2×D23—22no—
221C2×D23—22no—
222C2×D23—22no—
223C2×D23—22no—
224C2×D23—22no—
225C2×D23—41no—
226C2×D23—41no—
227C2×D23—41no—
228C2×D23—41no—
229C2×D23—41no—
230C2×D23—41no—
231C2×D23—41no—
232C2×D23—41no—
233C4×C23—22no—
234C4×C23—41no—
235C4×C23—41no—
236C4×C23—41no—
237C4×C23—41no—
238Dic23—22no—
239Dic23—22no—
240Dic23—22no—
241Dic23—41no—
242Dic23—41no—
243Dic23#12014yes—
244C22×#41—231no—
245C2×C22⋊Q8—231no—
246C2×#121—231no—
247C2×C4×D4—231no—
248#394—231no—
249#394D2311yescharacteristic, complemented
250#392—231no—
251#392—231no—
252#391—231no—
253#390—231no—
254#386—231no—
255#386—231no—
256#385—231no—
257#385—231no—
258#380—231no—
259C2×C23—42no—
260C2×C23—42no—
261C2×C23C2×#12011yescharacteristic
262C2×C23#52511yescharacteristic
263C2×C23#53711yescharacteristic
264C2×C23#54611yescharacteristic
265C2×C23#54711yescharacteristic
266C2×C23#54811yescharacteristic
267C2×C23#55011yescharacteristic
268D23—42no—
269D23—42no—
270C2×#41—464no—
271C2×#41—464no—
272C2×#41—464no—
273C22×Q8—231no—
274C22×D4—231no—
275C22×D4—231no—
276C22×D4—231no—
277C22×D4—231no—
278C22×D4—231no—
279C22×D4C2×D2311yescharacteristic
280C23×C4—231no—
281C23×C4—231no—
282C23×C4—231no—
283C23×C4—231no—
284C22⋊Q8—464no—
285#121—464no—
286C4×D4—464no—
287C2×C4⋊C4—231no—
288C2×C4⋊C4—231no—
289C2×C4⋊C4—231no—
290C2×C4⋊C4—231no—
291C2×C4⋊C4C2×D2311yescharacteristic
292C2×C22⋊C4—231no—
293C2×C22⋊C4—231no—
294C2×C22⋊C4—231no—
295C2×C22⋊C4—231no—
296C2×C22⋊C4—231no—
297C2×C22⋊C4—231no—
298C2×C22⋊C4—231no—
299C2×C22⋊C4—231no—
300C2×C22⋊C4C2×D2311yescharacteristic
301C2×C22⋊C4C2×D2311yescharacteristic
302C2×C22⋊C4C2×D2311yescharacteristic
303C2×C22⋊C4C2×D2311yescharacteristic
304C2×C42—231no—
305#95—231no—
306#95—231no—
307#95—231no—
308#95—231no—
309#95—231no—
310#95C2×D2311yescharacteristic
311C23#240711yescharacteristic, complemented, Sylow
312C24—231no—
313C24—231no—
314C24C22×D2311yescharacteristic
315C24C22×D2311yescharacteristic
316#41—4632no—
317C2×Q8—234no—
318C2×Q8—462no—
319C2×Q8—462no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—234no—
324C2×D4—462no—
325C2×D4—462no—
326C2×D4—462no—
327C2×D4—462no—
328C2×D4—462no—
329C2×D4—462no—
330C2×D4—462no—
331C2×D4—462no—
332C2×D4—462no—
333C2×D4—462no—
334C2×D4—462no—
335C2×D4—462no—
336C2×D4—921no—
337C2×D4—921no—
338C22×C4—231no—
339C22×C4—234no—
340C22×C4—231no—
341C22×C4—231no—
342C22×C4—231no—
343C22×C4—231no—
344C22×C4—231no—
345C22×C4—462no—
346C22×C4—462no—
347C22×C4—462no—
348C22×C4—462no—
349C22×C4—462no—
350C22×C4—462no—
351C22×C4—462no—
352C22×C4—462no—
353C22×C4—462no—
354C22×C4—462no—
355C22×C4—462no—
356C22×C4C22×D2311yescharacteristic
357C22×C4C22×D2311yescharacteristic
358C22×C4C22×D2311yescharacteristic
359C22×C4C22×D2311yescharacteristic
360C22×C4C22×D2311yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—462no—
363C4⋊C4—462no—
364C4⋊C4—462no—
365C4⋊C4—462no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—462no—
371C22⋊C4—462no—
372C22⋊C4—462no—
373C22⋊C4—462no—
374C22⋊C4—462no—
375C22⋊C4—462no—
376C22⋊C4—921no—
377C22⋊C4—921no—
378C42—462no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—232no—
392C23—232no—
393C23—461no—
394C23—461no—
395C23—461no—
396C23—461no—
397C23—461no—
398C23—461no—
399C23—461no—
400C23—461no—
401C23—461no—
402C23—461no—
403C23—461no—
404C23—461no—
405C23D4×D2312yes—
406C23D4×D2312yes—
407C23C23×D2311yescharacteristic
408Q8—468no—
409D4—28no—
410D4—468no—
411D4—468no—
412D4—468no—
413D4—924no—
414D4—924no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—232no—
430C2×C4—232no—
431C2×C4—232no—
432C2×C4—232no—
433C2×C4—232no—
434C2×C4—232no—
435C2×C4—468no—
436C2×C4—468no—
437C2×C4—468no—
438C2×C4—461no—
439C2×C4—461no—
440C2×C4—461no—
441C2×C4—461no—
442C2×C4—461no—
443C2×C4—461no—
444C2×C4—461no—
445C2×C4—461no—
446C2×C4—461no—
447C2×C4—461no—
448C2×C4—461no—
449C2×C4—461no—
450C2×C4—924no—
451C2×C4D4×D2312yes—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—462no—
467C22—462no—
468C22—462no—
469C22—462no—
470C22—462no—
471C22—462no—
472C22—921no—
473C22—921no—
474C22—921no—
475C22—921no—
476C22—921no—
477C22—921no—
478C22—921no—
479C22—921no—
480C22C2×D4×D2311yescharacteristic
481C22C2×D4×D2311yescharacteristic
482C22C2×D4×D2311yescharacteristic
483C22#1062515511yescharacteristic
484C22#41×D2311yescharacteristic
485C22#41×D2311yescharacteristic
486C22#1062516111yescharacteristic
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—234no—
493C4—462no—
494C4—462no—
495C4—462no—
496C4—921no—
497C4—921no—
498C2—42no—
499C2—42no—
500C2—922no—
501C2—922no—
502C2#120×D2311yescharacteristic
503C2#1309292511yescharacteristic
504C2#1309299411yescharacteristic
505C2#1309305611yescharacteristic
506C2#1309306511yescharacteristic
507C2#1309307511yescharacteristic
508C2#1309308511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#394C2
hypercenter#394D23
hypocenterC23#2407
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#67290C2semidirect
2. C2×#10625067C2semidirect
3. C2×#10625055C2semidirect
4. C2×(C22⋊D23)⋊C4C2semidirect
5. #13092332C2semidirect
6. #13092326C2semidirect
7. #13092325C2semidirect
8. #13092321C2semidirect
9. #13092317C2semidirect
10. #13092295C2semidirect
11. #13092294C2semidirect
12. #13092289C2semidirect
13. #13092288C2semidirect
14. #13092275C2semidirect
15. C23×#394C2semidirect
80. C2×C22⋊D23D4semidirect
86. C22×Dic23D4semidirect
105. C2×Dic46D4semidirect
152. C22×D23#41semidirect
192. C2×Dic23#41semidirect
249. #394D23semidirect
311. C23#2407semidirect

Maximal subgroups

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

subgroupindex
C22×#672902
C2×#106250672
C2×#106250552
C2×(C22⋊D23)⋊C42
#130923322
#130923262
#130923252
#130923212
#130923172
#130922952
#130922942
#130922892
#130922882
#130922752
C23×#3942
#240723