Finite Group Databasebeta

← #15682777

Subgroups — #15682777

Order 2944.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13093253C211yesmaximal, characteristic
2C2×#10625080C211yesmaximal, characteristic, complemented
3C2×#10625122C211yesmaximal, characteristic, complemented
4#13092993C211yesmaximal, characteristic, complemented
5#13092601C214yesmaximal, complemented
6#13092596C214yesmaximal, complemented
7#13092435C211yesmaximal, characteristic, complemented
8#13092399C211yesmaximal, characteristic, complemented
9C23×#421C211yesmaximal, characteristic, complemented
10C2×#41×C23C2211yescharacteristic
11#10625161—21no—
12#10625161C2212yes—
13#10625160—21no—
14#10625160C2212yes—
15#41×D23—22no—
16#41×D23—22no—
17#10625158—21no—
18#10625155—21no—
19#10625155—22no—
20C2×#67288—22no—
21C2×#67288C2211yescharacteristic
22C2×#67288C2211yescharacteristic
23C22×Dic46C2211yescharacteristic, complemented
24C2×M16×C23C2211yescharacteristic, complemented
25C23×#117C2211yescharacteristic, complemented
26#10625122—22no—
27#10625122C2214yescomplemented
28C2×#67275—21no—
29C2×#67273—21no—
30C2×#67267C2211yescharacteristic, complemented
31#10625080—22no—
32#10625080C2214yescomplemented
33C2×Dic92—21no—
34C2×#67263—21no—
35#10625068—22no—
36#10625057—22no—
37#10625046C2212yescomplemented
38#10625040C2212yescomplemented
39C23×#104C2214yescomplemented
40#10625006C2212yescomplemented
41#10625005C2212yescomplemented
42#10624987C2214yescomplemented
43#41×C23—21no—
44#41×C23—21no—
45#41×C23D412yes—
46#41×C23C2312yes—
47C2×Q8×C23D411yescharacteristic
48C2×D4×C23—21no—
49C2×D4×C23D411yescharacteristic
50C22×C4×C23—21no—
51C22×C4×C23C2311yescharacteristic
52C2×C22⋊D23—24no—
53C2×C22⋊D23—21no—
54C2×C22⋊D23D412yes—
55C22×Dic23—21no—
56#67292—22no—
57#67292—22no—
58Q8×D23—22no—
59Q8×D23—22no—
60#67290—24no—
61#67290—22no—
62#67290—22no—
63#67290—24no—
64D4×D23—24no—
65D4×D23—22no—
66D4×D23—24no—
67D4×D23—22no—
68#67288—24no—
69#67288—24no—
70#67288—21no—
71#67288—21no—
72#67288—21no—
73#67288—21no—
74#67288—21no—
75#67288—21no—
76#67288—22no—
77#67288—22no—
78#67288D412yes—
79#67288C2312yes—
80C2×D92—22no—
81C2×D92D411yescharacteristic
82C2×D92C2311yescharacteristic
83C2×C4×D23—24no—
84C2×C4×D23—21no—
85C2×C4×D23D412yes—
86C2×Dic46—22no—
87C2×Dic46—21no—
88C2×Dic46—21no—
89C2×Dic46—21no—
90C2×Dic46—21no—
91C2×Dic46—21no—
92C2×Dic46D411yescharacteristic
93C2×Dic46C2312yes—
94C2×Dic46C2311yescharacteristic
95M16×C23—21no—
96M16×C23C2312yes—
97C2×C8×C23—21no—
98C4⋊C4×C23—21no—
99C22⋊C4×C23—21no—
100C42×C23C2312yes—
101#67275—24no—
102#67273—24no—
103(C2×D23)⋊C4—24no—
104C4⋊Dic23—24no—
105#67267—21no—
106#67267C2312yes—
107C2×C23⋊C8—21no—
108Dic92—24no—
109#67263—24no—
110C23×C23—21no—
111C23×C23C2×D411yescharacteristic
112C22×D23—24no—
113C22×D23—21no—
114C22×D23C2×D412yes—
115Q8×C23—21no—
116Q8×C23C2×D412yes—
117D4×C23—22no—
118D4×C23—22no—
119D4×C23—21no—
120D4×C23C2×D412yes—
121C2×C4×C23—21no—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23—22no—
125C2×C4×C23C2×D411yescharacteristic
126C2×C4×C23C2×D411yescharacteristic
127C2×C4×C23C2×D411yescharacteristic
128C2×C4×C23C2×D411yescharacteristic
129C2×C4×C23C2×D411yescharacteristic
130C2×C4×C23C2411yescharacteristic
131C22⋊D23—24no—
132C22⋊D23—24no—
133C22⋊D23—22no—
134C22⋊D23—22no—
135C22⋊D23—22no—
136C22⋊D23—22no—
137C2×Dic23—24no—
138C2×Dic23—21no—
139C2×Dic23—22no—
140C2×Dic23—41no—
141C2×Dic23—41no—
142C2×Dic23C2×D412yes—
143D92—24no—
144D92—21no—
145D92—21no—
146D92—21no—
147D92C2×D412yes—
148C4×D23—24no—
149C4×D23—24no—
150C4×D23—22no—
151C4×D23—22no—
152C4×D23—22no—
153C4×D23—22no—
154Dic46—24no—
155Dic46—21no—
156Dic46—21no—
157Dic46—21no—
158Dic46—22no—
159Dic46—22no—
160Dic46—41no—
161Dic46—41no—
162Dic46C2×D412yes—
163C8×C23—22no—
164C23⋊C8—22no—
165#3022—231nomaximal, Sylow
166C22×C23—22no—
167C22×C23—41no—
168C22×C23—81no—
169C22×C23#12011yescharacteristic
170C22×C23#12011yescharacteristic
171C22×C23C22×D411yescharacteristic
172C2×D23—22no—
173C2×D23—22no—
174C2×D23—42no—
175C2×D23—81no—
176C2×D23—84no—
177C4×C23—22no—
178C4×C23—42no—
179C4×C23#12011yescharacteristic
180C4×C23#12011yescharacteristic
181C4×C23C22×D411yescharacteristic
182C4×C23C22×D411yescharacteristic
183Dic23—22no—
184Dic23—22no—
185Dic23—42no—
186#585—231no—
187C2×#137—231no—
188C2×#137—231no—
189#536—231no—
190#456—234no—
191#455—234no—
192#421—231no—
193#421D2311yescharacteristic, complemented
194#413—231no—
195C2×C23—21no—
196C2×C23—21no—
197C2×C23—21no—
198C2×C23—42no—
199C2×C23C2×#12011yescharacteristic
200D23—42no—
201D23—42no—
2022−1+4—232no—
2032−1+4—461no—
2042−1+4—461no—
205H(5,𝔽2)—232no—
206H(5,𝔽2)—462no—
207H(5,𝔽2)—461no—
208H(5,𝔽2)—461no—
209C2×#41—231no—
210C2×#41—231no—
211C2×#41—462no—
212C2×#41—462no—
213C2×#41—462no—
214C2×#41C2×D2311yescharacteristic
215C22×Q8—231no—
216#137—234no—
217#137—234no—
218#137—462no—
219#137—462no—
220C2×Q16—461no—
221C2×Q16—461no—
222C2×SD16—461no—
223C2×SD16—461no—
224C2×M16—231no—
225C2×M16C2×D2311yescharacteristic, complemented
226C4⋊Q8—232no—
227#124—232no—
228#123—462no—
229C22⋊Q8—462no—
230#117C2×D2311yescharacteristic, complemented
231#104—234no—
232#104C2×D2314yescomplemented
233#101—232no—
234#100—232no—
235C23#302211yescharacteristic, complemented, Sylow
236#41—21no—
237#41—21no—
238#41—232no—
239#41—232no—
240#41—462no—
241#41—462no—
242#41—464no—
243#41—464no—
244#41—464no—
245#41—464no—
246#41—462no—
247#41—462no—
248#41—462no—
249#41—462no—
250#41—461no—
251#41—461no—
252#41—461no—
253#41—461no—
254#41—461no—
255#41—461no—
256#41D9212yes—
257#41C22×D2312yes—
258C2×Q8—231no—
259C2×Q8—231no—
260C2×Q8—232no—
261C2×Q8—462no—
262C2×Q8—462no—
263C2×Q8—462no—
264C2×Q8—461no—
265C2×Q8—461no—
266C2×Q8—461no—
267C2×Q8—461no—
268C2×Q8—461no—
269C2×Q8D9211yescharacteristic
270C2×D4—21no—
271C2×D4—232no—
272C2×D4—231no—
273C2×D4—231no—
274C2×D4—462no—
275C2×D4—464no—
276C2×D4—464no—
277C2×D4—462no—
278C2×D4—464no—
279C2×D4—461no—
280C2×D4—462no—
281C2×D4D9211yescharacteristic
282C22×C4—21no—
283C22×C4—232no—
284C22×C4—464no—
285C22×C4—461no—
286C22×C4—461no—
287C22×C4C22×D2311yescharacteristic
288Q16—464no—
289Q16—464no—
290SD16—464no—
291SD16—464no—
292M16—21no—
293M16—232no—
294M16—461no—
295M16C22×D2312yes—
296C2×C8—21no—
297C2×C8—461no—
298C4⋊C4—21no—
299C4⋊C4—464no—
300C22⋊C4—21no—
301C22⋊C4—464no—
302C42C22×D2312yes—
303C23—21no—
304C23—232no—
305C23—464no—
306C23—461no—
307C23D4×D2311yescharacteristic
308Q8—21no—
309Q8—232no—
310Q8—464no—
311Q8—461no—
312Q8—461no—
313Q8—461no—
314Q8—462no—
315Q8—462no—
316Q8—921no—
317Q8—921no—
318Q8C2×D9212yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—232no—
323D4—464no—
324D4—464no—
325D4—464no—
326D4—462no—
327D4—462no—
328D4—462no—
329D4—462no—
330D4—461no—
331D4—461no—
332D4—461no—
333D4C2×D9212yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—232no—
339C2×C4—464no—
340C2×C4—464no—
341C2×C4—464no—
342C2×C4—461no—
343C2×C4—462no—
344C2×C4—462no—
345C2×C4—462no—
346C2×C4—462no—
347C2×C4—462no—
348C2×C4—921no—
349C2×C4—921no—
350C2×C4C2×D9211yescharacteristic
351C2×C4C2×D9211yescharacteristic
352C2×C4D4×D2311yescharacteristic
353C2×C4D4×D2311yescharacteristic
354C2×C4D4×D2311yescharacteristic
355C2×C4C23×D2311yescharacteristic
356C8—22no—
357C8—462no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—462no—
362C22—462no—
363C22—922no—
364C22—1841no—
365C22—1844no—
366C22#1062505311yescharacteristic
367C22#1062505311yescharacteristic
368C22C2×D4×D2311yescharacteristic
369C4—22no—
370C4—42no—
371C4—462no—
372C4—462no—
373C4—922no—
374C4#1062505311yescharacteristic
375C4#1062505311yescharacteristic
376C4C22×D9211yescharacteristic
377C4C2×D4×D2311yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—922no—
383C2—922no—
384C2C2×#1062505311yescharacteristic
subgroupG/H
centerC2C2×#10625053
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#421C2
socleC2×C23C2×#120
prime radical#13093253C2
hypercenter#421D23
hypocenterC23#3022
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#10625080C2semidirect
3. C2×#10625122C2semidirect
4. #13092993C2semidirect
5. #13092601C2semidirect
6. #13092596C2semidirect
7. #13092435C2semidirect
8. #13092399C2semidirect
9. C23×#421C2semidirect
23. C22×Dic46C22semidirect
24. C2×M16×C23C22semidirect
25. C23×#117C22semidirect
27. #10625122C22semidirect
30. C2×#67267C22semidirect
32. #10625080C22semidirect
37. #10625046C22semidirect
38. #10625040C22semidirect
39. C23×#104C22semidirect
40. #10625006C22semidirect
41. #10625005C22semidirect
42. #10624987C22semidirect
193. #421D23semidirect
225. C2×M16C2×D23semidirect
230. #117C2×D23semidirect
232. #104C2×D23semidirect
235. C23#3022semidirect

Maximal subgroups

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

subgroupindex
#130932532
C2×#106250802
C2×#106251222
#130929932
#130926012
#130925962
#130924352
#130923992
C23×#4212
#302223