Finite Group Databasebeta

← #49175517

Subgroups — #49175517

Order 37504.

502 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627711C211yesmaximal, characteristic, complemented
2C22×#17989811C211yesmaximal, characteristic, complemented
3#34037164C211yesmaximal, characteristic, complemented
4#34037162C211yesmaximal, characteristic, complemented
5#34037161C211yesmaximal, characteristic, complemented
6#34037160C211yesmaximal, characteristic, complemented
7#34037160C211yesmaximal, characteristic, complemented
8#34037158C211yesmaximal, characteristic, complemented
9#34037156C211yesmaximal, characteristic, complemented
10D4×#14556880C211yesmaximal, characteristic, complemented
11#34037135C211yesmaximal, characteristic, complemented
12#34037102C211yesmaximal, characteristic, complemented
13C2×C22⋊#14556880C211yesmaximal, characteristic, complemented
14#34037093C211yesmaximal, characteristic, complemented
15#34036635C211yesmaximal, characteristic
16#22627711—24no—
17C2×#17989812C2211yescharacteristic, complemented
18C2×#17989811—24no—
19C2×#17989811C2211yescharacteristic
20C2×#17989811C2211yescharacteristic
21C2×#17989811C2211yescharacteristic
22C2×#17989811C2211yescharacteristic
23C2×#17989811C2211yescharacteristic, complemented
24C2×D293⋊C8C2211yescharacteristic, complemented
25C22×#14556880C2211yescharacteristic
26C22×#14556880C2211yescharacteristic
27C22×#14556880C2211yescharacteristic
28C22×#14556880C2211yescharacteristic
29C22×#14556880C2211yescharacteristic, complemented
30#22627691C2211yescharacteristic, complemented
31#22627691C2211yescharacteristic, complemented
32C4⋊#14556880C2211yescharacteristic, complemented
33C4⋊#14556880C2211yescharacteristic, complemented
34#22627681C2211yescharacteristic, complemented
35#22627681C2211yescharacteristic, complemented
36C22⋊#14556880C2211yescharacteristic
37C22⋊#14556880C2211yescharacteristic
38C22⋊#14556880C2211yescharacteristic
39C22⋊#14556880C2211yescharacteristic
40C22⋊#14556880C2211yescharacteristic, complemented
41C22⋊#14556880C2211yescharacteristic, complemented
42#22627679C2211yescharacteristic, complemented
43#22627679C2211yescharacteristic, complemented
44C4×#14556880C2211yescharacteristic, complemented
45C4×#14556880C2211yescharacteristic, complemented
46C2×#17989801C2211yescharacteristic
47C23×Dic293C2211yescharacteristic
48#22627608C411yescharacteristic
49#22627605C411yescharacteristic
50#22627601C411yescharacteristic
51#22627601C411yescharacteristic
52#22627596C411yescharacteristic
53#22627596C411yescharacteristic
54#22627593C411yescharacteristic
55D4×Dic293C2211yescharacteristic
56D4×Dic293C2211yescharacteristic
57#22627584C2211yescharacteristic
58#22627584C2211yescharacteristic
59#22627580C2211yescharacteristic
60#120×C293C411yescharacteristic
61#17989812—22no—
62#17989811—22no—
63#17989811—22no—
64#17989811—22no—
65#17989811—22no—
66#17989811D414yescomplemented
67D293⋊C8—22no—
68C2×#14556880—22no—
69C2×#14556880—22no—
70C2×#14556880—22no—
71C2×#14556880—22no—
72C2×#14556880—22no—
73C2×#14556880C2311yescharacteristic
74C2×#14556880C2311yescharacteristic
75C2×#14556880C2311yescharacteristic
76C2×#14556880C2311yescharacteristic
77C2×#14556880C2311yescharacteristic
78C2×#14556880C2311yescharacteristic
79C2×#14556880C2311yescharacteristic
80C2×#14556880C2311yescharacteristic
81#17989801—22no—
82#17989801—22no—
83C2×Dic586C2×C411yescharacteristic
84C2×C22⋊D293C2×C411yescharacteristic
85C2×C22⋊D293C2×C411yescharacteristic
86C2×C22⋊D293C2×C411yescharacteristic
87C2×C22⋊D293C2×C411yescharacteristic
88C22×Dic293—22no—
89C22×Dic293—22no—
90C22×Dic293C2×C411yescharacteristic
91C22×Dic293C2×C411yescharacteristic
92C22×Dic293C2×C411yescharacteristic
93C22×Dic293C2311yescharacteristic
94C22×Dic293C2311yescharacteristic
95C22×Dic293C2311yescharacteristic
96C22×Dic293C2311yescharacteristic
97C2×C4×D293C2311yescharacteristic
98C4⋊Dic293C2×C411yescharacteristic
99C4⋊Dic293C2×C411yescharacteristic
100#17989781C2×C411yescharacteristic
101#17989781C2×C411yescharacteristic
102#17989781C2×C411yescharacteristic
103#17989781C2×C411yescharacteristic
104C22⋊Dic293C2×C411yescharacteristic
105C22⋊Dic293C2×C411yescharacteristic
106C22⋊Dic293C2×C411yescharacteristic
107C22⋊Dic293C2×C411yescharacteristic
108C22⋊Dic293C2×C411yescharacteristic
109#17989779C2×C411yescharacteristic
110#17989779C2×C411yescharacteristic
111C4×Dic293C2311yescharacteristic
112C4×Dic293C2311yescharacteristic
113C24×C293C2×C411yescharacteristic
114C2×D4×C293C2×C411yescharacteristic
115C2×D4×C293C2×C411yescharacteristic
116C2×D4×C293C2×C411yescharacteristic
117C22⋊C4×C293C2×C411yescharacteristic
118C22⋊C4×C293C2×C411yescharacteristic
119C22⋊C4×C293C2×C411yescharacteristic
120#14556880—21no—
121#14556880—21no—
122#14556880—21no—
123#14556880—21no—
124#14556880—21no—
125#14556880—21no—
126#14556880C2×D412yes—
127#14556880C2×D412yes—
128C22×D293C22×C411yescharacteristic
129Dic586—22no—
130C22⋊D293—22no—
131C22⋊D293—22no—
132C22⋊D293—22no—
133C22⋊D293—41no—
134C2×Dic293—22no—
135C2×Dic293—22no—
136C2×Dic293—22no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293—21no—
141C2×Dic293—21no—
142C2×Dic293—21no—
143C2×Dic293C22×C411yescharacteristic
144C2×Dic293C22×C411yescharacteristic
145C2×Dic293C22×C411yescharacteristic
146C2×Dic293C22×C411yescharacteristic
147C2×Dic293C22×C411yescharacteristic
148C2×Dic293C22×C411yescharacteristic
149C2×Dic293C2×D412yes—
150C2×Dic293#4112yes—
151C2×Dic293C2411yescharacteristic
152C4×D293—21no—
153C4×D293—21no—
154C23×C293—22no—
155C23×C293—22no—
156C23×C293—41no—
157C23×C293C22×C411yescharacteristic
158C23×C293C22×C411yescharacteristic
159C23×C293C22×C411yescharacteristic
160C23×C293C22×C411yescharacteristic
161D4×C293—22no—
162D4×C293—22no—
163D4×C293—41no—
164C2×C4×C293C22×C411yescharacteristic
165C2×C4×C293C22×C411yescharacteristic
166C2×C4×C293C22×C411yescharacteristic
167C22×C293—21no—
168C22×C293—21no—
169C22×C293—21no—
170C22×C293—21no—
171C22×C293—21no—
172C22×C293—21no—
173C22×C293—21no—
174C22×C293—21no—
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—44no—
178C22×C293C4×D412yes—
179C22×C293#13112yescomplemented
180C22×C293C23×C411yescharacteristic
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—21no—
184C4×C293—21no—
185C4×C293—21no—
186C4×C293—21no—
187Dic293—21no—
188Dic293—21no—
189Dic293—21no—
190Dic293—21no—
191Dic293—21no—
192Dic293C4×D412yes—
193Dic293C22×D411yescharacteristic
194Dic293C2×#4111yescharacteristic
195C2×C293—22no—
196C2×C293—22no—
197C2×C293—41no—
198C2×C293—41no—
199C2×C293C2×C4×D411yescharacteristic
200C2×C293C2×#13111yescharacteristic
201C2×C293#56811yescharacteristic
202D293—41no—
203C293#293311yescharacteristic, complemented, Sylow
204#2933—2931nomaximal, Sylow
205C2×#131—2931no—
206C22×M16—2931no—
207#525—2931no—
208#436—2931no—
209#436—2931no—
210#435—2931no—
211#435—2931no—
212#435—2931no—
213#435—2931no—
214C8×D4—2931no—
215C8×D4—2931no—
216#424—2931no—
217#408—2931no—
218C2×C22⋊C8—2931no—
219#405—2931no—
220C2×#41—2931no—
221C23×C4—2931no—
222#131—5864no—
223C2×M16—2931no—
224C2×M16—2931no—
225C2×M16—2931no—
226C2×M16—2931no—
227C2×M16—2931no—
228C2×M16—2931no—
229C2×M16—5864no—
230C22×C8—2931no—
231C22×C8—2931no—
232C22×C8—2931no—
233C22×C8—2931no—
234C22×C8—2931no—
235C22×C8—2931no—
236#123—2931no—
237#123—2931no—
238C22⋊Q8—2931no—
239C22⋊Q8—2931no—
240#121—2931no—
241#121—2931no—
242#120—2931no—
243#120#1192407811yescharacteristic
244C4×D4—2931no—
245C4×D4—2931no—
246C4×D4—2931no—
247C4×D4—2931no—
248#117—2931no—
249C4⋊C8—2931no—
250C4⋊C8—2931no—
251C4⋊C8—2931no—
252C4⋊C8—2931no—
253C22⋊C8—2931no—
254C22⋊C8—2931no—
255C22⋊C8—2931no—
256C22⋊C8—2931no—
257C22⋊C8—2931no—
258C22⋊C8—2931no—
259C22⋊C8—2931no—
260C22⋊C8—2931no—
261#97—2931no—
262#97—2931no—
263C4×C8—2931no—
264C4×C8—2931no—
265C24C2×#1192407811yescharacteristic
266#41—5862no—
267#41—5862no—
268C2×Q8—2931no—
269C2×D4—2931no—
270C2×D4—2931no—
271C2×D4—2931no—
272C2×D4—2931no—
273C2×D4C2×#1192407811yescharacteristic
274C2×D4C2×#1192407811yescharacteristic
275C2×D4C2×#1192407811yescharacteristic
276C22×C4—2931no—
277C22×C4—2931no—
278C22×C4—2931no—
279C22×C4—2931no—
280C22×C4—2931no—
281C22×C4—2931no—
282C22×C4—2931no—
283C22×C4—2931no—
284C22×C4—5862no—
285C22×C4—5862no—
286M16—2934no—
287M16—5862no—
288M16—5862no—
289M16—5862no—
290M16—5862no—
291M16—5862no—
292C2×C8—2931no—
293C2×C8—2931no—
294C2×C8—2931no—
295C2×C8—2931no—
296C2×C8—2931no—
297C2×C8—2931no—
298C2×C8—2931no—
299C2×C8—2931no—
300C2×C8—5862no—
301C2×C8—5862no—
302C2×C8—5862no—
303C2×C8—5862no—
304C2×C8—5862no—
305C2×C8—5862no—
306C4⋊C4—2931no—
307C4⋊C4—2931no—
308C4⋊C4—2931no—
309C4⋊C4—2931no—
310C4⋊C4—2931no—
311C4⋊C4—2931no—
312C22⋊C4—2931no—
313C22⋊C4—2931no—
314C22⋊C4—2931no—
315C22⋊C4—2931no—
316C22⋊C4—2931no—
317C22⋊C4—2931no—
318C22⋊C4—2931no—
319C22⋊C4C2×#1192407811yescharacteristic
320C22⋊C4C2×#1192407811yescharacteristic
321C22⋊C4C2×#1192407811yescharacteristic
322C42—2931no—
323C42—2931no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—2931no—
328C23C22×#1192407811yescharacteristic
329C23C22×#1192407811yescharacteristic
330C23C22×#1192407811yescharacteristic
331C23C22×#1192407811yescharacteristic
332Q8—5862no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—5862no—
337D4—5862no—
338D4—5862no—
339D4—11721no—
340C2×C4—2932no—
341C2×C4—2932no—
342C2×C4—2931no—
343C2×C4—2931no—
344C2×C4—2931no—
345C2×C4—2931no—
346C2×C4—2931no—
347C2×C4—2931no—
348C2×C4—2931no—
349C2×C4—5861no—
350C2×C4—5861no—
351C2×C4—5861no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—5861no—
356C2×C4—5862no—
357C2×C4—5862no—
358C2×C4—5861no—
359C2×C4—5862no—
360C2×C4C22×#1192407811yescharacteristic
361C2×C4C22×#1192407811yescharacteristic
362C2×C4C22×#1192407811yescharacteristic
363C8—2932no—
364C8—2932no—
365C8—5861no—
366C8—5861no—
367C8—5861no—
368C8—5861no—
369C8—5861no—
370C8—5861no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—5861no—
383C22—5861no—
384C22—5861no—
385C22D4×#1192407812yes—
386C22#2262771112yescomplemented
387C22C23×#1192407811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—2931no—
392C4—2931no—
393C4—2932no—
394C4—5861no—
395C4—5861no—
396C4—5861no—
397C4—5861no—
398C4—5861no—
399C2—22no—
400C2—22no—
401C2—41no—
402C2—41no—
403C2—11721no—
404C2C2×D4×#1192407811yescharacteristic
405C2C2×#2262771111yescharacteristic
406C2#3403720711yescharacteristic
subgroupG/H
center, FrattiniC22C23×#11924078
commutator, socleC22×C293C23×C4
Fitting#120×C293C4
prime radical#34036635C2
cosocleC2×Dic293C24
hypercenter#120#11924078
hypocenterC293#2933
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#22627711C2semidirect
2. C22×#17989811C2semidirect
3. #34037164C2semidirect
4. #34037162C2semidirect
5. #34037161C2semidirect
6. #34037160C2semidirect
7. #34037160C2semidirect
8. #34037158C2semidirect
9. #34037156C2semidirect
10. D4×#14556880C2semidirect
11. #34037135C2semidirect
12. #34037102C2semidirect
13. C2×C22⋊#14556880C2semidirect
14. #34037093C2semidirect
17. C2×#17989812C22semidirect
23. C2×#17989811C22semidirect
24. C2×D293⋊C8C22semidirect
29. C22×#14556880C22semidirect
30. #22627691C22semidirect
31. #22627691C22semidirect
32. C4⋊#14556880C22semidirect
33. C4⋊#14556880C22semidirect
34. #22627681C22semidirect
35. #22627681C22semidirect
40. C22⋊#14556880C22semidirect
41. C22⋊#14556880C22semidirect
42. #22627679C22semidirect
43. #22627679C22semidirect
44. C4×#14556880C22semidirect
45. C4×#14556880C22semidirect
66. #17989811D4semidirect
179. C22×C293#131semidirect
203. C293#2933semidirect
386. C22#22627711semidirect

Maximal subgroups

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

subgroupindex
C2×#226277112
C22×#179898112
#340371642
#340371622
#340371612
#340371602
#340371582
#340371562
D4×#145568802
#340371352
#340371022
C2×C22⋊#145568802
#340370932
#340366352
#2933293