Finite Group Databasebeta

← #15676095

Subgroups — #15676095

Order 2944.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625110C211yesmaximal, characteristic
2C2×C4×C22⋊D23C211yesmaximal, characteristic, complemented
3#13092308C211yesmaximal, characteristic, complemented
4#13092305C211yesmaximal, characteristic, complemented
5#13092301C211yesmaximal, characteristic, complemented
6#13092298C211yesmaximal, characteristic, complemented
7#13092297C211yesmaximal, characteristic, complemented
8#13092296C211yesmaximal, characteristic, complemented
9#13092294C211yesmaximal, characteristic, complemented
10#13092293C211yesmaximal, characteristic, complemented
11#13092290C211yesmaximal, characteristic, complemented
12#13092288C211yesmaximal, characteristic, complemented
13#13092275C211yesmaximal, characteristic, complemented
14C2×C23⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C23C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C23×C4×C23C2211yescharacteristic
18C23×C4×C23C2211yescharacteristic
19C22×C22⋊D23C2211yescharacteristic
20C22×C22⋊D23C2211yescharacteristic
21C23×Dic23C2211yescharacteristic
22C23×Dic23C2211yescharacteristic
23C22×C4×D23C2211yescharacteristic
24C4×D4×C23—24no—
25C2×C4⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C42×C23C2211yescharacteristic
29C2×C22⋊Dic23C2211yescharacteristic
30C2×C22⋊Dic23C2211yescharacteristic
31C2×C22⋊Dic23C2211yescharacteristic
32C2×C22⋊Dic23C2211yescharacteristic
33#10625110C418yescomplemented
34C4×C22⋊D23—24no—
35C2×(C2×D23)⋊C4C2211yescharacteristic
36C2×(C2×D23)⋊C4C2211yescharacteristic
37C2×(C2×D23)⋊C4C2211yescharacteristic
38C2×(C2×D23)⋊C4C2211yescharacteristic
39C2×Dic23⋊C4C2211yescharacteristic
40C2×Dic23⋊C4C2211yescharacteristic
41C2×Dic23⋊C4C2211yescharacteristic
42C2×Dic23⋊C4C2211yescharacteristic
43C2×C4×Dic23C2211yescharacteristic
44C23⋊#95C2211yescharacteristic
45C23⋊#95C2211yescharacteristic
46C23⋊#95C2211yescharacteristic
47C23⋊#95C2211yescharacteristic
48C23⋊#95C2211yescharacteristic
49C23⋊#95C2211yescharacteristic
50C23⋊#95C2211yescharacteristic
51C23⋊#95C2211yescharacteristic
52C23⋊#95C2211yescharacteristic
53C23⋊#95C2211yescharacteristic
54C24×C23C2311yescharacteristic
55C24×C23C2311yescharacteristic
56C23×D23C2311yescharacteristic
57C2×D4×C23—22no—
58C2×D4×C23—22no—
59C2×D4×C23C2×C414yescomplemented
60C22×C4×C23—22no—
61C22×C4×C23—41no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
64C22×C4×C23—41no—
65C22×C4×C23D414yescomplemented
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C22×C4×C23C2311yescharacteristic
70C22×C4×C23C2311yescharacteristic
71C2×C22⋊D23—22no—
72C2×C22⋊D23—22no—
73C2×C22⋊D23—41no—
74C2×C22⋊D23—41no—
75C2×C22⋊D23C2×C414yescomplemented
76C2×C22⋊D23C2×C414yescomplemented
77C22×Dic23—22no—
78C22×Dic23—41no—
79C22×Dic23—41no—
80C22×Dic23—41no—
81C22×Dic23—41no—
82C22×Dic23C2×C414yescomplemented
83C22×Dic23C2311yescharacteristic
84C22×Dic23C2311yescharacteristic
85C22×Dic23C2311yescharacteristic
86C22×Dic23C2311yescharacteristic
87C22×Dic23C2311yescharacteristic
88C22×Dic23C2311yescharacteristic
89C22×Dic23C2311yescharacteristic
90C2×C4×D23—22no—
91C2×C4×D23—41no—
92C2×C4×D23—41no—
93C4⋊C4×C23—22no—
94C22⋊C4×C23—22no—
95C22⋊C4×C23—22no—
96C42×C23—22no—
97C22⋊Dic23—22no—
98C22⋊Dic23—41no—
99C22⋊Dic23—41no—
100C22⋊Dic23C2×C414yescomplemented
101(C2×D23)⋊C4—22no—
102(C2×D23)⋊C4—41no—
103(C2×D23)⋊C4—41no—
104(C2×D23)⋊C4C2×C414yescomplemented
105Dic23⋊C4—22no—
106Dic23⋊C4—41no—
107Dic23⋊C4—41no—
108Dic23⋊C4C2×C414yescomplemented
109C4×Dic23—22no—
110C23×C23—21no—
111C23×C23—21no—
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—21no—
120C23×C23C22×C412yes—
121C23×C23C22×C412yes—
122C23×C23C2×D412yes—
123C23×C23#4112yes—
124C23×C23C2411yescharacteristic
125C22×D23—21no—
126C22×D23—21no—
127C22×D23—21no—
128C22×D23—21no—
129C22×D23—21no—
130C22×D23—21no—
131C22×D23C22×C412yes—
132D4×C23—28no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
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—44no—
146C2×C4×C23—44no—
147C2×C4×C23C22×C412yes—
148C2×C4×C23C2×D412yes—
149C2×C4×C23C2×D412yes—
150C22⋊D23—28no—
151C22⋊D23—44no—
152C2×Dic23—21no—
153C2×Dic23—21no—
154C2×Dic23—21no—
155C2×Dic23—21no—
156C2×Dic23—21no—
157C2×Dic23—21no—
158C2×Dic23—21no—
159C2×Dic23—21no—
160C2×Dic23—21no—
161C2×Dic23—21no—
162C2×Dic23—21no—
163C2×Dic23—21no—
164C2×Dic23—21no—
165C2×Dic23—21no—
166C2×Dic23—21no—
167C2×Dic23—21no—
168C2×Dic23—21no—
169C2×Dic23—21no—
170C2×Dic23—44no—
171C2×Dic23—44no—
172C2×Dic23C22×C412yes—
173C2×Dic23C22×C412yes—
174C2×Dic23C22×C412yes—
175C4×D23—44no—
176#2325—231nomaximal, Sylow
177C22×C23—22no—
178C22×C23—22no—
179C22×C23—22no—
180C22×C23—22no—
181C22×C23—22no—
182C22×C23—22no—
183C22×C23—22no—
184C22×C23—22no—
185C22×C23—22no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23C4×D414yes—
191C22×C23C23×C411yescharacteristic
192C22×C23C22×D411yescharacteristic
193C22×C23C2×#4111yescharacteristic
194C22×C23H(5,𝔽2)11yescharacteristic
195C22×C23H(5,𝔽2)11yescharacteristic
196C22×C23H(5,𝔽2)11yescharacteristic
197C22×C232−1+411yescharacteristic
198C2×D23—22no—
199C2×D23—22no—
200C2×D23—22no—
201C2×D23—41no—
202C2×D23—41no—
203C2×D23—41no—
204C2×D23—41no—
205C4×C23—22no—
206C4×C23—22no—
207C4×C23—22no—
208C4×C23—41no—
209C4×C23—41no—
210Dic23—22no—
211Dic23—22no—
212Dic23—22no—
213Dic23—41no—
214Dic23—41no—
215Dic23—41no—
216Dic23—41no—
217C2×#121—231no—
218C2×C4×D4—231no—
219C2×C4×D4D2311yescharacteristic, complemented
220#388—231no—
221#388—231no—
222#387—231no—
223#387—231no—
224#386—231no—
225#386—231no—
226#386—231no—
227#386—231no—
228#385—231no—
229#385—231no—
230#380—231no—
231C2×#95—231no—
232C2×C23—24no—
233C2×C23—42no—
234C2×C23C2×C4×D411yescharacteristic
235C2×C23#51811yescharacteristic
236C2×C23#52011yescharacteristic
237C2×C23#53411yescharacteristic
238C2×C23#53711yescharacteristic
239C2×C23#53811yescharacteristic
240C2×C23#53911yescharacteristic
241D23—42no—
242C22×D4—231no—
243C22×D4—231no—
244C22×D4C2×D2311yescharacteristic
245C23×C4—231no—
246C23×C4—231no—
247C23×C4—231no—
248C23×C4C2×D2311yescharacteristic
249C23×C4C2×D2311yescharacteristic
250#121—238no—
251C4×D4—24no—
252C4×D4—464no—
253C2×C4⋊C4—231no—
254C2×C4⋊C4—231no—
255C2×C4⋊C4—231no—
256C2×C4⋊C4—231no—
257C2×C4⋊C4C2×D2311yescharacteristic
258C2×C22⋊C4—231no—
259C2×C22⋊C4—231no—
260C2×C22⋊C4—231no—
261C2×C22⋊C4—231no—
262C2×C22⋊C4—231no—
263C2×C22⋊C4—231no—
264C2×C22⋊C4—231no—
265C2×C22⋊C4—231no—
266C2×C22⋊C4C2×D2311yescharacteristic
267C2×C22⋊C4C2×D2311yescharacteristic
268C2×C42—231no—
269C2×C42C2×D2311yescharacteristic
270#95—231no—
271#95—231no—
272#95—231no—
273#95—231no—
274#95—231no—
275#95—231no—
276#95—231no—
277#95—231no—
278#95—231no—
279#95—231no—
280C23#232511yescharacteristic, complemented, Sylow
281C24—231no—
282C24C22×D2311yescharacteristic
283C24C22×D2311yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—234no—
287C2×D4—234no—
288C2×D4—462no—
289C2×D4—462no—
290C2×D4—921no—
291C2×D4—921no—
292C2×D4C4×D2314yescomplemented
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—231no—
299C22×C4—231no—
300C22×C4—231no—
301C22×C4—234no—
302C22×C4—231no—
303C22×C4—231no—
304C22×C4—231no—
305C22×C4—231no—
306C22×C4—462no—
307C22×C4—462no—
308C22×C4—921no—
309C22×C4—921no—
310C22×C4—921no—
311C22×C4—921no—
312C22×C4—921no—
313C22×C4—921no—
314C22×C4C22⋊D2314yescomplemented
315C22×C4C22×D2311yescharacteristic
316C22×C4C22×D2311yescharacteristic
317C22×C4C22×D2311yescharacteristic
318C22×C4C22×D2311yescharacteristic
319C22×C4C22×D2311yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—234no—
322C4⋊C4—462no—
323C4⋊C4—921no—
324C4⋊C4—921no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—234no—
328C22⋊C4—234no—
329C22⋊C4—462no—
330C22⋊C4—462no—
331C22⋊C4—921no—
332C22⋊C4—921no—
333C22⋊C4—921no—
334C22⋊C4—921no—
335C42—22no—
336C42—462no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—232no—
348C23—461no—
349C23—461no—
350C23—461no—
351C23—461no—
352C23—461no—
353C23—461no—
354C23C2×C4×D2312yes—
355C23C2×C4×D2312yes—
356C23#6728812yes—
357C23C2×C22⋊D2312yes—
358C23C23×D2311yescharacteristic
359D4—28no—
360D4—468no—
361D4—924no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—232no—
377C2×C4—232no—
378C2×C4—232no—
379C2×C4—461no—
380C2×C4—461no—
381C2×C4—461no—
382C2×C4—461no—
383C2×C4—461no—
384C2×C4—461no—
385C2×C4—461no—
386C2×C4—461no—
387C2×C4—461no—
388C2×C4—461no—
389C2×C4—461no—
390C2×C4—461no—
391C2×C4—461no—
392C2×C4—461no—
393C2×C4—461no—
394C2×C4—461no—
395C2×C4—461no—
396C2×C4—461no—
397C2×C4—924no—
398C2×C4—924no—
399C2×C4—924no—
400C2×C4C2×C4×D2312yes—
401C2×C4C2×C22⋊D2312yes—
402C2×C4C2×C22⋊D2312yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—462no—
417C22—462no—
418C22—462no—
419C22—921no—
420C22—921no—
421C22—921no—
422C22—921no—
423C22C4×C22⋊D2314yes—
424C22C22×C4×D2311yescharacteristic
425C22C2×#6728811yescharacteristic
426C22#1062515511yescharacteristic
427C22#1062515511yescharacteristic
428C22#1062516011yescharacteristic
429C22#1062516111yescharacteristic
430C22C22×C22⋊D2311yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—462no—
437C4—462no—
438C4—462no—
439C4—921no—
440C4—921no—
441C4—921no—
442C4—921no—
443C2—24no—
444C2—42no—
445C2—922no—
446C2C2×C4×C22⋊D2311yescharacteristic
447C2#1309288911yescharacteristic
448C2#1309289611yescharacteristic
449C2#1309297611yescharacteristic
450C2#1309299511yescharacteristic
451C2#1309300011yescharacteristic
452C2#1309300911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC2×C4×D4×C23C2
socle, cosocleC23×C23C24
prime radicalC2×#10625110C2
hypercenterC2×C4×D4D23
hypocenterC23#2325
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D23C2semidirect
3. #13092308C2semidirect
4. #13092305C2semidirect
5. #13092301C2semidirect
6. #13092298C2semidirect
7. #13092297C2semidirect
8. #13092296C2semidirect
9. #13092294C2semidirect
10. #13092293C2semidirect
11. #13092290C2semidirect
12. #13092288C2semidirect
13. #13092275C2semidirect
14. C2×C23⋊#95C2semidirect
15. C2×C4×D4×C23C2semidirect
33. #10625110C4semidirect
59. C2×D4×C23C2×C4semidirect
65. C22×C4×C23D4semidirect
75. C2×C22⋊D23C2×C4semidirect
76. C2×C22⋊D23C2×C4semidirect
82. C22×Dic23C2×C4semidirect
100. C22⋊Dic23C2×C4semidirect
104. (C2×D23)⋊C4C2×C4semidirect
108. Dic23⋊C4C2×C4semidirect
219. C2×C4×D4D23semidirect
280. C23#2325semidirect
292. C2×D4C4×D23semidirect
314. C22×C4C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251102
C2×C4×C22⋊D232
#130923082
#130923052
#130923012
#130922982
#130922972
#130922962
#130922942
#130922932
#130922902
#130922882
#130922752
C2×C23⋊#952
C2×C4×D4×C232
#232523