Finite Group Databasebeta

← #15676118

Subgroups — #15676118

Order 2944.

740 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625111C211yesmaximal, characteristic
2C2×C4×C22⋊D23C211yesmaximal, characteristic, complemented
3C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
4#13092317C211yesmaximal, characteristic, complemented
5#13092317C211yesmaximal, characteristic, complemented
6#13092297C211yesmaximal, characteristic, complemented
7#13092296C211yesmaximal, characteristic, complemented
8#13092295C211yesmaximal, characteristic, complemented
9#13092295C211yesmaximal, characteristic, complemented
10#13092294C211yesmaximal, characteristic, complemented
11#13092293C211yesmaximal, characteristic, complemented
12#13092293C211yesmaximal, characteristic, complemented
13#13092281C211yesmaximal, characteristic, complemented
14#13092262C211yesmaximal, characteristic, complemented
15C23×#386C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C23×C4×C23C2211yescharacteristic
18C22×C22⋊D23C2211yescharacteristic
19C22×C22⋊D23C2211yescharacteristic
20C22×C22⋊D23C2211yescharacteristic
21C22×C22⋊D23C2211yescharacteristic
22C23×Dic23C2211yescharacteristic
23C22×D92C2211yescharacteristic
24C22×C4×D23C2211yescharacteristic
25C22×C4×D23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C22⋊C4×C23C2211yescharacteristic
29C2×C22⋊Dic23C2211yescharacteristic
30C2×C22⋊Dic23C2211yescharacteristic
31C2×C22⋊Dic23C2211yescharacteristic
32#10625111C418yescomplemented
33C4×C22⋊D23—24no—
34C2×(C2×D23)⋊C4C2211yescharacteristic
35C2×(C2×D23)⋊C4C2211yescharacteristic
36C2×(C2×D23)⋊C4C2211yescharacteristic
37C2×(C2×D23)⋊C4C2211yescharacteristic
38C2×(C2×D23)⋊C4C2211yescharacteristic
39C2×(C2×D23)⋊C4C2211yescharacteristic
40C2×Dic23⋊C4C2211yescharacteristic
41C2×Dic23⋊C4C2211yescharacteristic
42C2×C4×Dic23C2211yescharacteristic
43C2×C4×Dic23C2211yescharacteristic
44C2×C4×Dic23C2211yescharacteristic
45(C22⋊D23)⋊C4—24no—
46C23×#95C2211yescharacteristic
47C23×#95C2211yescharacteristic
48C23⋊#95C2211yescharacteristic
49C23⋊#95C2211yescharacteristic
50C23⋊#95C2211yescharacteristic
51C23⋊#95C2211yescharacteristic
52C23⋊#95C2211yescharacteristic
53C23⋊#95C2211yescharacteristic
54C24×C23C2311yescharacteristic
55C24×C23C2311yescharacteristic
56C23×D23C2311yescharacteristic
57C23×D23C2311yescharacteristic
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23C2×C414yescomplemented
61C22×C4×C23—22no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
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—41no—
74C2×C22⋊D23—41no—
75C2×C22⋊D23—41no—
76C2×C22⋊D23—41no—
77C2×C22⋊D23C2×C414yescomplemented
78C2×C22⋊D23C2×C414yescomplemented
79C2×C22⋊D23C2×C414yescomplemented
80C2×C22⋊D23C2×C414yescomplemented
81C22×Dic23—22no—
82C22×Dic23—41no—
83C22×Dic23—41no—
84C22×Dic23C2311yescharacteristic
85C22×Dic23C2311yescharacteristic
86C22×Dic23C2311yescharacteristic
87C22×Dic23C2311yescharacteristic
88C22×Dic23C2311yescharacteristic
89C22×Dic23C2311yescharacteristic
90C2×D92—41no—
91C2×D92—41no—
92C2×D92C2×C414yescomplemented
93C2×C4×D23—22no—
94C2×C4×D23—22no—
95C2×C4×D23—41no—
96C2×C4×D23—41no—
97C2×C4×D23—41no—
98C2×C4×D23—41no—
99C22⋊C4×C23—22no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—41no—
102C22⋊Dic23—22no—
103C22⋊Dic23—41no—
104C22⋊Dic23—41no—
105(C2×D23)⋊C4—22no—
106(C2×D23)⋊C4—22no—
107(C2×D23)⋊C4—41no—
108(C2×D23)⋊C4—41no—
109(C2×D23)⋊C4—41no—
110(C2×D23)⋊C4—41no—
111Dic23⋊C4—22no—
112Dic23⋊C4—22no—
113C4×Dic23—22no—
114C4×Dic23C2×C414yescomplemented
115C4×Dic23D414yescomplemented
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—21no—
120C23×C23—21no—
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23C22×C412yes—
129C23×C23C22×C412yes—
130C23×C23C2411yescharacteristic
131C22×D23—21no—
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×D23C22×C412yes—
144C22×D23C22×C412yes—
145D4×C23—44no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—44no—
160C2×C4×C23C22×C412yes—
161C2×C4×C23C2×D412yes—
162C22⋊D23—28no—
163C22⋊D23—28no—
164C22⋊D23—44no—
165C22⋊D23—44no—
166C2×Dic23—21no—
167C2×Dic23—21no—
168C2×Dic23—21no—
169C2×Dic23—21no—
170C2×Dic23—21no—
171C2×Dic23—21no—
172C2×Dic23—21no—
173C2×Dic23—21no—
174C2×Dic23—21no—
175C2×Dic23—21no—
176C2×Dic23—21no—
177C2×Dic23—21no—
178C2×Dic23—21no—
179C2×Dic23—44no—
180C2×Dic23C22×C412yes—
181C2×Dic23C22×C412yes—
182C2×Dic23C2×D412yes—
183C2×Dic23C2×D412yes—
184C2×Dic23#4112yes—
185D92—44no—
186C4×D23—44no—
187C4×D23—44no—
188#2328—231nomaximal, Sylow
189C22×C23—22no—
190C22×C23—22no—
191C22×C23—22no—
192C22×C23—22no—
193C22×C23—22no—
194C22×C23—22no—
195C22×C23—41no—
196C22×C23—41no—
197C22×C23—41no—
198C22×C23—41no—
199C22×C23—41no—
200C22×C23—41no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23C23×C411yescharacteristic
204C22×C23C22×D411yescharacteristic
205C22×C23C2×#4111yescharacteristic
206C22×C23H(5,𝔽2)11yescharacteristic
207C22×C23H(5,𝔽2)11yescharacteristic
208C22×C23H(5,𝔽2)11yescharacteristic
209C22×C23H(5,𝔽2)11yescharacteristic
210C2×D23—22no—
211C2×D23—22no—
212C2×D23—22no—
213C2×D23—22no—
214C2×D23—22no—
215C2×D23—22no—
216C2×D23—41no—
217C2×D23—41no—
218C2×D23—41no—
219C2×D23—41no—
220C2×D23—41no—
221C2×D23—41no—
222C2×D23—41no—
223C2×D23—41no—
224C4×C23—22no—
225C4×C23—22no—
226C4×C23—41no—
227C4×C23—41no—
228C4×C23—41no—
229Dic23—22no—
230Dic23—22no—
231Dic23—41no—
232Dic23—41no—
233Dic23—41no—
234Dic23C4×D414yes—
235C2×#127—231no—
236C2×C4×D4—231no—
237C2×C4×D4—231no—
238#390—231no—
239#390—231no—
240#386—231no—
241#386—231no—
242#386—231no—
243#386—231no—
244#386—231no—
245#386—231no—
246#386—231no—
247#386D2311yescharacteristic, complemented
248#382—231no—
249#376—231no—
250C2×C23—42no—
251C2×C23—42no—
252C2×C23C2×C4×D411yescharacteristic
253C2×C23#51811yescharacteristic
254C2×C23#51811yescharacteristic
255C2×C23#53511yescharacteristic
256C2×C23#53511yescharacteristic
257C2×C23#54011yescharacteristic
258C2×C23#54011yescharacteristic
259D23—42no—
260D23—42no—
261C22×D4—231no—
262C22×D4—231no—
263C22×D4—231no—
264C22×D4—231no—
265C22×D4—231no—
266C22×D4C2×D2311yescharacteristic
267C23×C4—231no—
268C23×C4—231no—
269C23×C4—231no—
270C23×C4C2×D2311yescharacteristic
271#127—238no—
272C4×D4—464no—
273C4×D4—464no—
274C2×C4⋊C4—231no—
275C2×C4⋊C4—231no—
276C2×C22⋊C4—231no—
277C2×C22⋊C4—231no—
278C2×C22⋊C4—231no—
279C2×C22⋊C4—231no—
280C2×C22⋊C4—231no—
281C2×C22⋊C4—231no—
282C2×C22⋊C4—231no—
283C2×C22⋊C4—231no—
284C2×C22⋊C4—231no—
285C2×C22⋊C4C2×D2311yescharacteristic
286C2×C22⋊C4C2×D2311yescharacteristic
287C2×C22⋊C4C2×D2311yescharacteristic
288C2×C42—231no—
289C2×C42—231no—
290C2×C42—231no—
291#95—231no—
292#95—231no—
293#95—231no—
294#95—231no—
295#95—231no—
296#95—231no—
297#95C2×D2311yescharacteristic
298#95C2×D2311yescharacteristic
299C23#232811yescharacteristic, complemented, Sylow
300C24—231no—
301C24—231no—
302C24C22×D2311yescharacteristic
303C24C22×D2311yescharacteristic
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—234no—
307C2×D4—234no—
308C2×D4—234no—
309C2×D4—234no—
310C2×D4—234no—
311C2×D4—462no—
312C2×D4—462no—
313C2×D4—462no—
314C2×D4—462no—
315C2×D4—921no—
316C2×D4—921no—
317C2×D4—921no—
318C2×D4—921no—
319C2×D4—921no—
320C2×D4—921no—
321C2×D4C4×D2314yescomplemented
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—231no—
326C22×C4—231no—
327C22×C4—231no—
328C22×C4—231no—
329C22×C4—231no—
330C22×C4—231no—
331C22×C4—462no—
332C22×C4—462no—
333C22×C4—462no—
334C22×C4—921no—
335C22×C4—921no—
336C22×C4—921no—
337C22×C4—921no—
338C22×C4—921no—
339C22×C4—921no—
340C22×C4C22×D2311yescharacteristic
341C22×C4C22×D2311yescharacteristic
342C22×C4C22×D2311yescharacteristic
343C22×C4C22×D2311yescharacteristic
344C22×C4C22×D2311yescharacteristic
345C4⋊C4—462no—
346C4⋊C4—462no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—462no—
351C22⋊C4—462no—
352C22⋊C4—462no—
353C22⋊C4—921no—
354C22⋊C4—921no—
355C22⋊C4—921no—
356C22⋊C4—921no—
357C22⋊C4—921no—
358C22⋊C4—921no—
359C42—234no—
360C42—234no—
361C42—462no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—232no—
375C23—232no—
376C23—461no—
377C23—461no—
378C23—461no—
379C23—461no—
380C23—461no—
381C23—461no—
382C23—461no—
383C23—461no—
384C23—461no—
385C23—461no—
386C23—461no—
387C23—461no—
388C23C2×C4×D2312yes—
389C23C2×C4×D2312yes—
390C23C23×D2311yescharacteristic
391D4—44no—
392D4—468no—
393D4—468no—
394D4—924no—
395D4—924no—
396D4—924no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—232no—
412C2×C4—232no—
413C2×C4—232no—
414C2×C4—232no—
415C2×C4—232no—
416C2×C4—461no—
417C2×C4—461no—
418C2×C4—461no—
419C2×C4—461no—
420C2×C4—461no—
421C2×C4—461no—
422C2×C4—461no—
423C2×C4—461no—
424C2×C4—461no—
425C2×C4—461no—
426C2×C4—461no—
427C2×C4—461no—
428C2×C4—461no—
429C2×C4—924no—
430C2×C4—924no—
431C2×C4—924no—
432C2×C4C2×C4×D2312yes—
433C2×C4D4×D2312yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—462no—
449C22—462no—
450C22—462no—
451C22—462no—
452C22—462no—
453C22—462no—
454C22—921no—
455C22—921no—
456C22—921no—
457C22—921no—
458C22—921no—
459C22—921no—
460C22—921no—
461C22—921no—
462C22C22×C4×D2311yescharacteristic
463C22C2×D4×D2311yescharacteristic
464C22#1062515511yescharacteristic
465C22#1062515511yescharacteristic
466C22#1062515511yescharacteristic
467C22#41×D2311yescharacteristic
468C22#1062516011yescharacteristic
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—234no—
475C4—462no—
476C4—462no—
477C4—921no—
478C4—921no—
479C4—921no—
480C2—42no—
481C2—42no—
482C2—922no—
483C2—922no—
484C2C4×D4×D2311yescharacteristic
485C2#1309288911yescharacteristic
486C2#1309289011yescharacteristic
487C2#1309298211yescharacteristic
488C2#1309298311yescharacteristic
489C2#1309301711yescharacteristic
490C2#1309301911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#386C2
socle, cosocleC23×C23C24
prime radicalC2×#10625111C2
hypercenter#386D23
hypocenterC23#2328
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D23C2semidirect
3. C2×(C22⋊D23)⋊C4C2semidirect
4. #13092317C2semidirect
5. #13092317C2semidirect
6. #13092297C2semidirect
7. #13092296C2semidirect
8. #13092295C2semidirect
9. #13092295C2semidirect
10. #13092294C2semidirect
11. #13092293C2semidirect
12. #13092293C2semidirect
13. #13092281C2semidirect
14. #13092262C2semidirect
15. C23×#386C2semidirect
32. #10625111C4semidirect
60. C2×D4×C23C2×C4semidirect
77. C2×C22⋊D23C2×C4semidirect
78. C2×C22⋊D23C2×C4semidirect
79. C2×C22⋊D23C2×C4semidirect
80. C2×C22⋊D23C2×C4semidirect
92. C2×D92C2×C4semidirect
114. C4×Dic23C2×C4semidirect
115. C4×Dic23D4semidirect
247. #386D23semidirect
299. C23#2328semidirect
321. C2×D4C4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251112
C2×C4×C22⋊D232
C2×(C22⋊D23)⋊C42
#130923172
#130922972
#130922962
#130922952
#130922942
#130922932
#130922812
#130922622
C23×#3862
#232823