Finite Group Databasebeta

← #15684908

Subgroups — #15684908

Order 2944.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625113C211yesmaximal, characteristic, complemented
2C22×#67274C211yesmaximal, characteristic, complemented
3#13093063C211yesmaximal, characteristic
4#13092809C211yesmaximal, characteristic, complemented
5#13092665C211yesmaximal, characteristic, complemented
6#13092656C211yesmaximal, characteristic, complemented
7#13092639C211yesmaximal, characteristic, complemented
8#13092619C211yesmaximal, characteristic, complemented
9#13092577C211yesmaximal, characteristic, complemented
10#13092573C211yesmaximal, characteristic, complemented
11#13092567C211yesmaximal, characteristic, complemented
12#13092556C211yesmaximal, characteristic, complemented
13#13092515C211yesmaximal, characteristic, complemented
14#13092489C211yesmaximal, characteristic, complemented
15D4×Q8×C23C211yesmaximal, characteristic, complemented
16C22×Q8×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic, complemented
18C2×#67288C2211yescharacteristic
19C22×D92C2211yescharacteristic
20C23×C4⋊Q8—21no—
21C23×C4⋊Q8C2211yescharacteristic, complemented
22C23×C22⋊Q8—21no—
23C23×C22⋊Q8—21no—
24C23×C22⋊Q8C2211yescharacteristic
25C23×C22⋊Q8C2211yescharacteristic, complemented
26C4×Q8×C23C2211yescharacteristic, complemented
27C4×D4×C23—21no—
28C4×D4×C23C2211yescharacteristic
29C2×#67275C2211yescharacteristic, complemented
30C2×#67275C2211yescharacteristic, complemented
31C2×#67275C2211yescharacteristic, complemented
32#10625113—24no—
33C2×#67274—24no—
34C2×#67274C2211yescharacteristic
35C2×#67274C2211yescharacteristic
36C2×#67274C2211yescharacteristic
37C2×#67274C2211yescharacteristic
38C2×#67274C2211yescharacteristic
39C2×#67274C2211yescharacteristic
40#10625108—21no—
41#10625101C2211yescharacteristic
42#10625095C2211yescharacteristic
43C2×#67267C2211yescharacteristic, complemented
44C22×C23⋊C8C2211yescharacteristic
45#10625065—21no—
46#10625055—21no—
47C22⋊C4×D23—21no—
48#10625046C2211yescharacteristic
49C4×D92C2211yescharacteristic
50#10625017C2211yescharacteristic, complemented
51#10625017C2211yescharacteristic, complemented
52#10625017C2211yescharacteristic, complemented
53C22⋊(C23⋊C8)C2211yescharacteristic
54C22⋊(C23⋊C8)C2211yescharacteristic, complemented
55#10624992C2211yescharacteristic, complemented
56#10624991C2211yescharacteristic
57#10624991C2211yescharacteristic
58#10624989C2211yescharacteristic, complemented
59C4⋊(C23⋊C8)C2211yescharacteristic, complemented
60#10624985C2211yescharacteristic, complemented
61C23×D23—21no—
62C2×Q8×C23—21no—
63C2×Q8×C23—21no—
64C2×Q8×C23—22no—
65C2×Q8×C23—22no—
66C2×Q8×C23—41no—
67C2×Q8×C23—41no—
68C2×Q8×C23C2311yescharacteristic
69C2×Q8×C23C2311yescharacteristic
70C2×Q8×C23C2311yescharacteristic
71C2×D4×C23D411yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23C2311yescharacteristic
75C22×C4×C23C2311yescharacteristic
76C2×C22⋊D23—21no—
77C2×C22⋊D23—21no—
78#67288—22no—
79#67288—41no—
80C2×D92—22no—
81C2×D92—21no—
82C2×D92—41no—
83C2×D92C2311yescharacteristic
84C2×D92C2311yescharacteristic
85C2×C4×D23—21no—
86C2×C4×D23—21no—
87C2×Dic46C2311yescharacteristic
88C4⋊C4×C23—21no—
89C4⋊C4×C23—21no—
90C4⋊C4×C23—21no—
91C4⋊C4×C23—21no—
92C4⋊C4×C23—21no—
93C4⋊C4×C23D411yescharacteristic
94C4⋊C4×C23C2311yescharacteristic
95C22⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23D411yescharacteristic
98C22⋊C4×C23D411yescharacteristic
99C42×C23—21no—
100C42×C23C2311yescharacteristic
101C22⋊Dic23—21no—
102#67275—22no—
103#67275—22no—
104#67275—41no—
105#67274—22no—
106#67274—22no—
107#67274—22no—
108#67274—22no—
109#67274—41no—
110#67274D414yescomplemented
111(C2×D23)⋊C4—21no—
112(C2×D23)⋊C4—21no—
113(C2×D23)⋊C4—21no—
114(C2×D23)⋊C4—21no—
115C4⋊Dic23C2311yescharacteristic
116Dic23⋊C4—21no—
117#67267—22no—
118C2×C23⋊C8—22no—
119C2×C23⋊C8C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121C2×C23⋊C8C2311yescharacteristic
122C2×C23⋊C8C2311yescharacteristic
123C23×C23C2×D411yescharacteristic
124C23×C23C2×D411yescharacteristic
125C22×D23—21no—
126C22×D23—21no—
127C22×D23—42no—
128C22×D23—42no—
129C22×D23—81no—
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23—21no—
133Q8×C23—21no—
134Q8×C23—21no—
135Q8×C23—42no—
136Q8×C23—42no—
137Q8×C23C2×D412yes—
138D4×C23—22no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23C2×D411yescharacteristic
150C2×C4×C23C2×D411yescharacteristic
151C2×C4×C23C2×D411yescharacteristic
152C2×C4×C23C2×D411yescharacteristic
153C2×C4×C23C2411yescharacteristic
154C22⋊D23—42no—
155C22⋊D23—81no—
156C2×Dic23—21no—
157C2×Dic23—21no—
158D92—21no—
159D92—21no—
160D92—21no—
161D92—42no—
162D92C2×D412yes—
163C4×D23—42no—
164C4×D23—81no—
165Dic46—21no—
166Dic46—21no—
167C23⋊C8—21no—
168C23⋊C8—21no—
169C23⋊C8—21no—
170C23⋊C8C2×D412yes—
171#3266—231nomaximal, Sylow
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23#13712yescomplemented
178C22×C23C22×D411yescharacteristic
179C2×D23—22no—
180C2×D23—41no—
181C2×D23—41no—
182C2×D23—41no—
183C2×D23—41no—
184C2×D23—41no—
185C2×D23—84no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—22no—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—41no—
192C4×C23—41no—
193C4×C23—41no—
194C4×C23C22×D411yescharacteristic
195C4×C23H(5,𝔽2)11yescharacteristic
196Dic23—41no—
197Dic23—41no—
198C2×#137—231no—
199C22×SD16—231no—
200D4×Q8D2311yescharacteristic, complemented
201#546—231no—
202#497—231no—
203#468—231no—
204#467—231no—
205#464—231no—
206#461—231no—
207#452—231no—
208#451—231no—
209#450—231no—
210#448—231no—
211#440—231no—
212#435—231no—
213C2×C23—22no—
214C2×C23—41no—
215C2×C23D4211yescharacteristic
216C2×C23C2×#13711yescharacteristic
217C2×C23#57711yescharacteristic
218D23—42no—
219D23—81no—
220C2×#41—231no—
221C22×Q8C2×D2311yescharacteristic
222C22×Q8C2×D2311yescharacteristic, complemented
223C22×D4—231no—
224#137—464no—
225C2×Q16—231no—
226C2×Q16—231no—
227C2×Q16—231no—
228C2×SD16—231no—
229C2×SD16—231no—
230C2×SD16—231no—
231C2×SD16—231no—
232C2×SD16—231no—
233C2×SD16—231no—
234C2×SD16—464no—
235C2×M16—231no—
236C22×C8—231no—
237C4⋊Q8—21no—
238C4⋊Q8C2×D2311yescharacteristic, complemented
239#124—231no—
240#123—461no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—231no—
244C22⋊Q8C2×D2311yescharacteristic
245C22⋊Q8C2×D2311yescharacteristic, complemented
246#121—231no—
247#121—461no—
248#120—461no—
249C4×Q8C2×D2311yescharacteristic, complemented
250C4×D4—21no—
251C4×D4—231no—
252C4×D4C2×D2311yescharacteristic
253C2×C22⋊C4—461no—
254#107—231no—
255C4⋊C8—231no—
256#103—231no—
257#103—231no—
258#103—231no—
259#103—231no—
260#102—231no—
261#102—231no—
262C22⋊C8—231no—
263C22⋊C8—231no—
264#97—231no—
265C23#326611yescharacteristic, complemented, Sylow
266C24—461no—
267#41—462no—
268#41—921no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—231no—
276C2×Q8C22×D2311yescharacteristic
277C2×Q8C22×D2311yescharacteristic
278C2×Q8C22×D2311yescharacteristic
279C2×D4—231no—
280C2×D4—231no—
281C2×D4—462no—
282C2×D4—461no—
283C2×D4—461no—
284C2×D4—461no—
285C2×D4—921no—
286C2×D4C22⋊D2311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—461no—
290C22×C4—461no—
291C22×C4C22×D2311yescharacteristic
292C22×C4C22×D2311yescharacteristic
293Q16—462no—
294Q16—462no—
295Q16—921no—
296SD16—234no—
297SD16—462no—
298SD16—462no—
299SD16—462no—
300SD16—462no—
301SD16—921no—
302M16—462no—
303C2×C8—231no—
304C2×C8—231no—
305C2×C8—231no—
306C2×C8—231no—
307C2×C8—462no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—231no—
314C4⋊C4—461no—
315C4⋊C4C22⋊D2311yescharacteristic
316C4⋊C4C22×D2311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—461no—
320C22⋊C4—461no—
321C22⋊C4—461no—
322C22⋊C4—461no—
323C22⋊C4—461no—
324C22⋊C4C22⋊D2311yescharacteristic
325C22⋊C4C22⋊D2311yescharacteristic
326C42—21no—
327C42C22×D2311yescharacteristic
328C23—461no—
329C23—461no—
330C23—922no—
331C23—922no—
332C23—1841no—
333C23C2×C22⋊D2311yescharacteristic
334C23C2×C22⋊D2311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—461no—
343Q8—461no—
344Q8D4×D2312yes—
345D4—22no—
346D4—232no—
347D4—461no—
348D4—461no—
349D4—461no—
350D4—922no—
351D4—922no—
352D4—1841no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—461no—
364C2×C4—461no—
365C2×C4—922no—
366C2×C4—1841no—
367C2×C4C2×C22⋊D2311yescharacteristic
368C2×C4C2×C22⋊D2311yescharacteristic
369C2×C4C2×C22⋊D2311yescharacteristic
370C2×C4C2×C22⋊D2311yescharacteristic
371C2×C4C23×D2311yescharacteristic
372C8—232no—
373C8—461no—
374C8—461no—
375C8—461no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—462no—
382C22—921no—
383C22—921no—
384C22—921no—
385C22—921no—
386C22—921no—
387C22—1844no—
388C22#1062511312yescomplemented
389C22C22×C22⋊D2311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—921no—
399C4—921no—
400C4C2×D4×D2311yescharacteristic
401C4#1062515511yescharacteristic
402C2—22no—
403C2—41no—
404C2—922no—
405C2—1841no—
406C2D4×C22⋊D2311yescharacteristic
407C2C2×#1062511311yescharacteristic
408C2#1309322511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingD4×Q8×C23C2
socleC22×C23C22×D4
prime radical#13093063C2
hypercenterD4×Q8D23
hypocenterC23#3266
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10625113C2semidirect
2. C22×#67274C2semidirect
4. #13092809C2semidirect
5. #13092665C2semidirect
6. #13092656C2semidirect
7. #13092639C2semidirect
8. #13092619C2semidirect
9. #13092577C2semidirect
10. #13092573C2semidirect
11. #13092567C2semidirect
12. #13092556C2semidirect
13. #13092515C2semidirect
14. #13092489C2semidirect
15. D4×Q8×C23C2semidirect
17. C22×Q8×C23C22semidirect
21. C23×C4⋊Q8C22semidirect
25. C23×C22⋊Q8C22semidirect
26. C4×Q8×C23C22semidirect
29. C2×#67275C22semidirect
30. C2×#67275C22semidirect
31. C2×#67275C22semidirect
43. C2×#67267C22semidirect
50. #10625017C22semidirect
51. #10625017C22semidirect
52. #10625017C22semidirect
54. C22⋊(C23⋊C8)C22semidirect
55. #10624992C22semidirect
58. #10624989C22semidirect
59. C4⋊(C23⋊C8)C22semidirect
60. #10624985C22semidirect
110. #67274D4semidirect
177. C22×C23#137semidirect
200. D4×Q8D23semidirect
222. C22×Q8C2×D23semidirect
238. C4⋊Q8C2×D23semidirect
245. C22⋊Q8C2×D23semidirect
249. C4×Q8C2×D23semidirect
265. C23#3266semidirect
388. C22#10625113semidirect

Maximal subgroups

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

subgroupindex
C2×#106251132
C22×#672742
#130930632
#130928092
#130926652
#130926562
#130926392
#130926192
#130925772
#130925732
#130925672
#130925562
#130925152
#130924892
D4×Q8×C232
#326623