Finite Group Databasebeta

← #15676741

Subgroups — #15676741

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×#67288C211yesmaximal, characteristic, complemented
2C2×#10625067C211yesmaximal, characteristic, complemented
3C2×#10625055C211yesmaximal, characteristic, complemented
4C2×#10625064C211yesmaximal, characteristic, complemented
5#13092334C211yesmaximal, characteristic, complemented
6#13092332C211yesmaximal, characteristic, complemented
7#13092327C211yesmaximal, characteristic, complemented
8#13092326C211yesmaximal, characteristic, complemented
9#13092321C211yesmaximal, characteristic, complemented
10#13092317C211yesmaximal, characteristic, complemented
11#13092296C211yesmaximal, characteristic, complemented
12#13092295C211yesmaximal, characteristic, complemented
13#13092288C211yesmaximal, characteristic, complemented
14#13092276C211yesmaximal, characteristic, complemented
15C23×#385C211yesmaximal, characteristic, complemented
16C23×C4×C23C2211yescharacteristic
17C22×C22⋊D23C2211yescharacteristic
18C22×C22⋊D23C2211yescharacteristic
19C22×C22⋊D23C2211yescharacteristic
20C2×#67288—24no—
21C2×#67288—24no—
22C2×#67288—24no—
23C22×D92C2211yescharacteristic
24C22×D92C2211yescharacteristic
25C22×D92C2211yescharacteristic
26C22×C4×D23C2211yescharacteristic
27C22×C4×D23C2211yescharacteristic
28C22×C4×D23C2211yescharacteristic
29C22×Dic46C2211yescharacteristic
30C2×C4⋊C4×C23C2211yescharacteristic
31C2×C4⋊C4×C23C2211yescharacteristic
32C2×C22⋊C4×C23C2211yescharacteristic
33C2×C22⋊C4×C23C2211yescharacteristic
34C2×C22⋊Dic23C2211yescharacteristic
35C2×(C2×D23)⋊C4C2211yescharacteristic
36C2×(C2×D23)⋊C4C2211yescharacteristic
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×(C2×D23)⋊C4C2211yescharacteristic
44C2×Dic23⋊C4C2211yescharacteristic
45C2×Dic23⋊C4C2211yescharacteristic
46C2×Dic23⋊C4C2211yescharacteristic
47C2×C4×Dic23C2211yescharacteristic
48#10625067—24no—
49#10625064—24no—
50#10625055—24no—
51C23×#95C2211yescharacteristic
52C23×#95C2211yescharacteristic
53C23⋊#95C2211yescharacteristic
54C23⋊#95C2211yescharacteristic
55C23⋊#95C2211yescharacteristic
56C23⋊#95C2211yescharacteristic
57C24×C23C2311yescharacteristic
58C23×D23C2311yescharacteristic
59C23×D23C2311yescharacteristic
60C23×D23C2311yescharacteristic
61C22×C4×C23—22no—
62C22×C4×C23—22no—
63C22×C4×C23—22no—
64C22×C4×C23C2311yescharacteristic
65C22×C4×C23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C22×C4×C23C2311yescharacteristic
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—41no—
78C2×C22⋊D23D414yescomplemented
79C22×Dic23C2311yescharacteristic
80C22×Dic23C2311yescharacteristic
81C22×Dic23C2311yescharacteristic
82C22×Dic23C2311yescharacteristic
83C22×Dic23C2311yescharacteristic
84#67288—232no—
85C2×D92—22no—
86C2×D92—22no—
87C2×D92—22no—
88C2×D92—22no—
89C2×D92—22no—
90C2×D92—22no—
91C2×D92—22no—
92C2×D92—22no—
93C2×D92—41no—
94C2×C4×D23—22no—
95C2×C4×D23—22no—
96C2×C4×D23—22no—
97C2×C4×D23—22no—
98C2×C4×D23—22no—
99C2×C4×D23—22no—
100C2×C4×D23—22no—
101C2×C4×D23—22no—
102C2×C4×D23D414yescomplemented
103C2×Dic46—22no—
104C2×Dic46—22no—
105C2×Dic46D414yescomplemented
106C4⋊C4×C23—22no—
107C4⋊C4×C23—22no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—22no—
110C22⋊Dic23—41no—
111(C2×D23)⋊C4—22no—
112(C2×D23)⋊C4—22no—
113(C2×D23)⋊C4—22no—
114(C2×D23)⋊C4—22no—
115(C2×D23)⋊C4—22no—
116(C2×D23)⋊C4—22no—
117(C2×D23)⋊C4—41no—
118(C2×D23)⋊C4—41no—
119(C2×D23)⋊C4—41no—
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×C23C2×D412yes—
131C23×C23C2411yescharacteristic
132C22×D23—21no—
133C22×D23—21no—
134C22×D23—21no—
135C22×D23—21no—
136C22×D23—21no—
137C22×D23—21no—
138C22×D23—21no—
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×D23C2×D412yes—
151C22×D23C2×D412yes—
152C22×D23#4112yescomplemented
153C2×C4×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×C23—21no—
169C2×C4×C23C2×D412yes—
170C2×C4×C23C2×D412yes—
171C2×C4×C23#4112yescomplemented
172C22⋊D23—28no—
173C22⋊D23—28no—
174C22⋊D23—44no—
175C2×Dic23—21no—
176C2×Dic23—21no—
177C2×Dic23—21no—
178C2×Dic23—21no—
179C2×Dic23—21no—
180C2×Dic23—21no—
181C2×Dic23—21no—
182C2×Dic23—21no—
183C2×Dic23—21no—
184C2×Dic23—21no—
185C2×Dic23—21no—
186C2×Dic23C2×D412yes—
187C2×Dic23C2×D412yes—
188C2×Dic23C2×D412yes—
189C2×Dic23C2×D412yes—
190D92—28no—
191D92—28no—
192D92—44no—
193C4×D23—28no—
194C4×D23—28no—
195C4×D23—44no—
196Dic46—28no—
197#2407—231nomaximal, Sylow
198C22×C23—22no—
199C22×C23—22no—
200C22×C23—22no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23—41no—
204C22×C23—41no—
205C22×C23C22×D411yescharacteristic
206C22×C23C22×D411yescharacteristic
207C22×C23C22×D411yescharacteristic
208C22×C23C2×#4111yescharacteristic
209C22×C23C2×#4111yescharacteristic
210C22×C23H(5,𝔽2)11yescharacteristic
211C22×C232−1+411yescharacteristic
212C2×D23—22no—
213C2×D23—22no—
214C2×D23—22no—
215C2×D23—22no—
216C2×D23—22no—
217C2×D23—22no—
218C2×D23—22no—
219C2×D23—22no—
220C2×D23—22no—
221C2×D23—41no—
222C2×D23—41no—
223C2×D23—41no—
224C2×D23—41no—
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—22no—
235C4×C23—22no—
236C4×C23—41no—
237C4×C23—41no—
238C4×C23—41no—
239Dic23—22no—
240Dic23—41no—
241Dic23—41no—
242Dic23—41no—
243Dic23#12014yes—
244C22×#41—231no—
245C2×C22⋊Q8—231no—
246C2×#121—231no—
247C2×C4×D4—231no—
248#394—231no—
249#394—231no—
250#392—231no—
251#392—231no—
252#391—231no—
253#390—231no—
254#386—231no—
255#386—231no—
256#385—231no—
257#385D2311yescharacteristic, complemented
258#380—231no—
259C2×C23—42no—
260C2×C23C2×#12011yescharacteristic
261C2×C23#52511yescharacteristic
262C2×C23#53711yescharacteristic
263C2×C23#54611yescharacteristic
264C2×C23#54711yescharacteristic
265C2×C23#54811yescharacteristic
266C2×C23#55011yescharacteristic
267D23—42no—
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×D4—231no—
280C23×C4—231no—
281C23×C4—231no—
282C23×C4—231no—
283C23×C4C2×D2311yescharacteristic
284C22⋊Q8—464no—
285#121—464no—
286C4×D4—464no—
287C2×C4⋊C4—231no—
288C2×C4⋊C4—231no—
289C2×C4⋊C4—231no—
290C2×C4⋊C4C2×D2311yescharacteristic
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⋊C4—231no—
301C2×C22⋊C4—231no—
302C2×C22⋊C4C2×D2311yescharacteristic
303C2×C22⋊C4C2×D2311yescharacteristic
304C2×C42—231no—
305#95—231no—
306#95—231no—
307#95—231no—
308#95—231no—
309#95C2×D2311yescharacteristic
310#95C2×D2311yescharacteristic
311C23#240711yescharacteristic, complemented, Sylow
312C24—231no—
313C24—231no—
314C24—231no—
315C24C22×D2311yescharacteristic
316#41—4632no—
317C2×Q8—234no—
318C2×Q8—462no—
319C2×Q8—462no—
320C2×D4—234no—
321C2×D4—462no—
322C2×D4—462no—
323C2×D4—462no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—231no—
342C22×C4—231no—
343C22×C4—234no—
344C22×C4—231no—
345C22×C4—231no—
346C22×C4—231no—
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×C4C22×D2311yescharacteristic
356C22×C4C22×D2311yescharacteristic
357C22×C4C22×D2311yescharacteristic
358C22×C4C22×D2311yescharacteristic
359C22×C4C22×D2311yescharacteristic
360C22×C4C22×D2311yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—462no—
364C4⋊C4—462no—
365C4⋊C4—462no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—462no—
369C22⋊C4—462no—
370C22⋊C4—462no—
371C22⋊C4—462no—
372C22⋊C4—462no—
373C22⋊C4—462no—
374C22⋊C4—921no—
375C22⋊C4—921no—
376C22⋊C4—921no—
377C22⋊C4—921no—
378C42—462no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—232no—
386C23—232no—
387C23—232no—
388C23—461no—
389C23—461no—
390C23—461no—
391C23—461no—
392C23—461no—
393C23—461no—
394C23—461no—
395C23—461no—
396C23—461no—
397C23—461no—
398C23—461no—
399C23—461no—
400C23—461no—
401C23—461no—
402C23—461no—
403C23—461no—
404C23—461no—
405C23—461no—
406C23D4×D2312yes—
407C23C23×D2311yescharacteristic
408Q8—468no—
409D4—468no—
410D4—468no—
411D4—468no—
412D4—468no—
413D4—924no—
414D4—924no—
415C2×C4—28no—
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—21no—
430C2×C4—21no—
431C2×C4—232no—
432C2×C4—232no—
433C2×C4—232no—
434C2×C4—232no—
435C2×C4—468no—
436C2×C4—468no—
437C2×C4—461no—
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—924no—
449C2×C4#6728812yescomplemented
450C2×C4D4×D2312yes—
451C2×C4D4×D2312yes—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—462no—
460C22—462no—
461C22—462no—
462C22—462no—
463C22—462no—
464C22—462no—
465C22—462no—
466C22—462no—
467C22—462no—
468C22—921no—
469C22—921no—
470C22—921no—
471C22—921no—
472C22—921no—
473C22—921no—
474C22—921no—
475C22—921no—
476C22—921no—
477C22—921no—
478C22—921no—
479C22—921no—
480C22C2×#6728811yescharacteristic
481C22C2×D4×D2311yescharacteristic
482C22C2×D4×D2311yescharacteristic
483C22C2×D4×D2311yescharacteristic
484C22#1062515811yescharacteristic
485C22#41×D2311yescharacteristic
486C22#1062516011yescharacteristic
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—234no—
494C4—462no—
495C4—921no—
496C4—921no—
497C4—921no—
498C2—42no—
499C2—922no—
500C2—922no—
501C2—922no—
502C2#120×D2311yescharacteristic
503C2#1309292611yescharacteristic
504C2#1309299711yescharacteristic
505C2#1309305911yescharacteristic
506C2#1309306811yescharacteristic
507C2#1309307711yescharacteristic
508C2#1309308811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#385C2
hypercenter#385D23
hypocenterC23#2407
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#67288C2semidirect
2. C2×#10625067C2semidirect
3. C2×#10625055C2semidirect
4. C2×#10625064C2semidirect
5. #13092334C2semidirect
6. #13092332C2semidirect
7. #13092327C2semidirect
8. #13092326C2semidirect
9. #13092321C2semidirect
10. #13092317C2semidirect
11. #13092296C2semidirect
12. #13092295C2semidirect
13. #13092288C2semidirect
14. #13092276C2semidirect
15. C23×#385C2semidirect
78. C2×C22⋊D23D4semidirect
102. C2×C4×D23D4semidirect
105. C2×Dic46D4semidirect
152. C22×D23#41semidirect
171. C2×C4×C23#41semidirect
257. #385D23semidirect
311. C23#2407semidirect
449. C2×C4#67288semidirect

Maximal subgroups

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

subgroupindex
C22×#672882
C2×#106250672
C2×#106250552
C2×#106250642
#130923342
#130923322
#130923272
#130923262
#130923212
#130923172
#130922962
#130922952
#130922882
#130922762
C23×#3852
#240723