Finite Group Databasebeta

← #49172375

Subgroups — #49172375

Order 37504.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036861C213yesmaximal, complemented
2#34036855C211yesmaximal, characteristic, complemented
3#34036850C213yesmaximal, complemented
4#34036831C213yesmaximal, complemented
5#34036825C213yesmaximal, complemented
6#34036798C211yesmaximal, characteristic, complemented
7#34036790C211yesmaximal, characteristic, complemented
8#34036674C213yesmaximal, complemented
9#34036665C213yesmaximal, complemented
10#34036658C213yesmaximal, complemented
11#41×Dic293C211yesmaximal, characteristic, complemented
12C4×#17989803C211yesmaximal, characteristic, complemented
13C4×#17989801C213yesmaximal, complemented
14C2×Q8×Dic293C211yesmaximal, characteristic, complemented
15#548×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2213yes—
18#22627623C2213yes—
19#22627620C2213yes—
20#22627619C2216yes—
21#22627617C2216yes—
22#22627616C2213yes—
23#22627615C2213yes—
24#22627613C2213yes—
25#22627610C2216yes—
26#22627608C2216yes—
27#22627606C2216yes—
28#22627600C2213yes—
29#22627598C2213yes—
30#22627598C2213yes—
31#22627596C2216yes—
32#22627595C2213yes—
33Q8×Dic293C2211yescharacteristic
34Q8×Dic293C2211yescharacteristic
35Q8×Dic293C2213yes—
36Q8×Dic293C2211yescharacteristic
37Q8×Dic293C2213yes—
38C4×Dic586C2213yes—
39#22627589C2213yes—
40C4×D1172C2213yes—
41D4×Dic293C2213yes—
42D4×Dic293C2213yes—
43D4×Dic293C2213yes—
44#22627586C2213yes—
45C4×C22⋊D293C2213yes—
46C4×C22⋊D293C2213yes—
47#22627584C2213yes—
48#22627584C2213yes—
49#22627584C2216yes—
50#22627583C2213yes—
51#22627582C2213yes—
52#22627582C2213yes—
53#22627580C2211yescharacteristic
54#22627580C2213yes—
55C2×C4⋊Dic293C2213yes—
56C2×C4×Dic293C2213yes—
57C2×C4×Dic293C2213yes—
58C42×D293C2213yes—
59C2×#41×C293C2211yescharacteristic
60C22×Q8×C293C2211yescharacteristic
61#124×C293C2213yes—
62C22⋊Q8×C293C2213yes—
63#121×C293C2213yes—
64C4×Q8×C293C2211yescharacteristic
65C4×D4×C293C2213yes—
66#17989803—24no—
67#17989801—212no—
68C2×Dic586C2313yes—
69C2×D1172C2313yes—
70C2×C22⋊D293C2313yes—
71C2×C22⋊D293C2313yes—
72C2×C22⋊D293C2316yes—
73C22×Dic293C2313yes—
74C22×Dic293C2313yes—
75C22×Dic293C2313yes—
76C22×Dic293C2311yescharacteristic
77C2×C4×D293C2313yes—
78C2×C4×D293C2313yes—
79C2×C4×D293C2313yes—
80C4⋊Dic293C2313yes—
81C4⋊Dic293C2313yes—
82C4⋊Dic293C2313yes—
83C4⋊Dic293C2313yes—
84C4⋊Dic293C2316yes—
85#17989781C2311yescharacteristic
86#17989781C2313yes—
87#17989781C2316yes—
88#17989781C2311yescharacteristic
89#17989781C2313yes—
90C22⋊Dic293C2311yescharacteristic
91C22⋊Dic293C2313yes—
92C22⋊Dic293C2316yes—
93#17989779C2313yes—
94#17989779C2313yes—
95#17989779C2316yes—
96#17989779C2316yes—
97C4×Dic293C2313yes—
98C4×Dic293C2313yes—
99C4×Dic293C2313yes—
100C4×Dic293C2313yes—
101C4×Dic293C2313yes—
102C4×Dic293C2313yes—
103C4×Dic293C2313yes—
104C4×Dic293C2311yescharacteristic
105C4×Dic293C2311yescharacteristic
106#41×C293—24no—
107C2×Q8×C293—24no—
108C2×Q8×C293C2313yes—
109C2×Q8×C293C2311yescharacteristic
110C2×D4×C293C2313yes—
111C2×D4×C293C2313yes—
112C22×C4×C293C2313yes—
113C22×C4×C293C2313yes—
114C4⋊C4×C293C2313yes—
115C4⋊C4×C293C2313yes—
116C22⋊C4×C293C2311yescharacteristic
117C22⋊C4×C293C2313yes—
118C22⋊C4×C293C2316yes—
119C42×C293C2313yes—
120C22×D293C2413yes—
121Dic586—26no—
122D1172—26no—
123C22⋊D293—26no—
124C22⋊D293—26no—
125C22⋊D293—46no—
126C2×Dic293—26no—
127C2×Dic293—43no—
128C2×Dic293—43no—
129C2×Dic293#4114yescomplemented
130C2×Dic293C2411yescharacteristic
131C2×Dic293C2413yes—
132C2×Dic293C2411yescharacteristic
133C2×Dic293C2413yes—
134C2×Dic293C2413yes—
135C2×Dic293C2411yescharacteristic
136C2×Dic293C2411yescharacteristic
137C4×D293—26no—
138C4×D293—26no—
139C4×D293—43no—
140C23×C293C2413yes—
141C23×C293C2411yescharacteristic
142Q8×C293—26no—
143Q8×C293#4114yescomplemented
144D4×C293—26no—
145D4×C293—26no—
146C2×C4×C293—26no—
147C2×C4×C293—26no—
148C2×C4×C293C2411yescharacteristic
149C2×C4×C293C2413yes—
150C2×C4×C293C2413yes—
151C2×C4×C293C2411yescharacteristic
152C2×C4×C293C2413yes—
153C22×C293—21no—
154C22×C293—21no—
155C22×C293—23no—
156C22×C293—23no—
157C22×C293—23no—
158C22×C293C2×#4112yes—
159C22×C293C2511yescharacteristic
160C2×D293—23no—
161C2×D293—23no—
162C2×D293—23no—
163C4×C293—21no—
164C4×C293—23no—
165C4×C293—23no—
166C4×C293—21no—
167C4×C293C2×#4116yes—
168Dic293—21no—
169Dic293—23no—
170Dic293—23no—
171Dic293—21no—
172Dic293—23no—
173Dic293C2×#4112yes—
174Dic293C2×#4112yes—
175C2×C293—22no—
176C2×C293—43no—
177C2×C293C22×#4111yescharacteristic
178C2×C293C22×#4111yescharacteristic
179C2×C293#58511yescharacteristic
180D293—43no—
181C293#348011yescharacteristic, complemented, Sylow
182#3480—2931nomaximal, Sylow
183#559—2933no—
184#557—2931no—
185#556—2933no—
186#553—2933no—
187#553—2933no—
188#550—2931no—
189#548—2931no—
190#548D29311yescharacteristic, complemented
191#532—2933no—
192#529—2933no—
193#529—2933no—
194C4×#41—2931no—
195C4×#41—2931no—
196C4×#41—2933no—
197C2×C4×Q8—2931no—
198C2×#41—2931no—
199C2×#41—2933no—
200C2×#41C2×D29311yescharacteristic
201C22×Q8C2×D29311yescharacteristic
202C4⋊Q8—2933no—
203#127—2933no—
204#126—2936no—
205#126—2936no—
206#125—2933no—
207#125—2933no—
208#124—2933no—
209#124—2936no—
210#124C2×D29313yes—
211#123—2936no—
212#123—2936no—
213C22⋊Q8—2933no—
214C22⋊Q8C2×D29313yes—
215#121—2933no—
216#121—2936no—
217#121—2933no—
218#121—2933no—
219#121C2×D29313yes—
220C4×Q8—2931no—
221C4×Q8—2933no—
222C4×Q8—2931no—
223C4×Q8—2933no—
224C4×Q8—2931no—
225C4×Q8—2933no—
226C4×Q8—2933no—
227C4×Q8C2×D29311yescharacteristic
228C4×D4—2933no—
229C4×D4—2933no—
230C4×D4—2933no—
231C4×D4—2933no—
232C4×D4—2933no—
233C4×D4—2933no—
234C4×D4—2933no—
235C4×D4—2933no—
236C4×D4—2936no—
237C4×D4—2933no—
238C4×D4C2×D29313yes—
239#117—2933no—
240#117—2933no—
241#117—2933no—
242#117—2931no—
243#117—2933no—
244C2×C4⋊C4—2933no—
245C2×C42—2933no—
246C2×C42—2933no—
247C2×C42—2933no—
248#41—24no—
249#41—58612no—
250#41—5864no—
251C2×Q8—24no—
252C2×Q8—2933no—
253C2×Q8C22×D29313yes—
254C2×Q8C22×D29311yescharacteristic
255C2×D4—2933no—
256C2×D4—2933no—
257C2×D4—2936no—
258C2×D4—2933no—
259C2×D4C22×D29313yes—
260C2×D4C22×D29313yes—
261C22×C4—2931no—
262C22×C4—2933no—
263C22×C4—2933no—
264C22×C4—2933no—
265C22×C4—2933no—
266C22×C4—2933no—
267C22×C4—2933no—
268C22×C4C22×D29313yes—
269C22×C4C22×D29313yes—
270C4⋊C4—2933no—
271C4⋊C4—2933no—
272C4⋊C4—2936no—
273C4⋊C4—2931no—
274C4⋊C4—2931no—
275C4⋊C4—2933no—
276C4⋊C4—2933no—
277C4⋊C4—2933no—
278C4⋊C4—2933no—
279C4⋊C4—2936no—
280C4⋊C4C22×D29313yes—
281C4⋊C4C22×D29313yes—
282C22⋊C4—2931no—
283C22⋊C4—2933no—
284C22⋊C4—2936no—
285C22⋊C4—2933no—
286C22⋊C4—2936no—
287C22⋊C4—2933no—
288C22⋊C4—2936no—
289C22⋊C4C22×D29311yescharacteristic
290C22⋊C4C22×D29313yes—
291C22⋊C4C22×D29316yes—
292C42—2931no—
293C42—2931no—
294C42—2933no—
295C42—2933no—
296C42—2933no—
297C42—2933no—
298C42—2933no—
299C42—2933no—
300C42—2933no—
301C42C22×D29313yes—
302C23—2933no—
303C23C23×D29313yes—
304C23C23×D29311yescharacteristic
305Q8—26no—
306Q8—5866no—
307Q8#1798980114yescomplemented
308D4—26no—
309D4—26no—
310D4—5866no—
311D4—5866no—
312D4—5866no—
313D4—11726no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2934no—
317C2×C4—2931no—
318C2×C4—2931no—
319C2×C4—2931no—
320C2×C4—2931no—
321C2×C4—2933no—
322C2×C4—2933no—
323C2×C4—2933no—
324C2×C4—5866no—
325C2×C4—5866no—
326C2×C4—5866no—
327C2×C4—11723no—
328C2×C4—11723no—
329C2×C4—11723no—
330C2×C4C23×D29311yescharacteristic
331C2×C4C23×D29313yes—
332C2×C4C23×D29311yescharacteristic
333C2×C4C23×D29313yes—
334C2×C4C23×D29313yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—5863no—
341C22—5863no—
342C22—5863no—
343C22#41×D29312yes—
344C22C24×D29311yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—2932no—
350C4—2932no—
351C4—5863no—
352C4—5861no—
353C4—5863no—
354C4—5861no—
355C4—5863no—
356C4C2×#1798980116yes—
357C2—22no—
358C2—43no—
359C2—11723no—
360C2C2×#41×D29311yescharacteristic
361C2C22×#1798980111yescharacteristic
362C2#3403696611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D293
commutator, socle, cosocleC22×C293C25
Fitting#548×C293C2
hypercenter#548D293
hypocenterC293#3480
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #34036861C2semidirect
2. #34036855C2semidirect
3. #34036850C2semidirect
4. #34036831C2semidirect
5. #34036825C2semidirect
6. #34036798C2semidirect
7. #34036790C2semidirect
8. #34036674C2semidirect
9. #34036665C2semidirect
10. #34036658C2semidirect
11. #41×Dic293C2semidirect
12. C4×#17989803C2semidirect
13. C4×#17989801C2semidirect
14. C2×Q8×Dic293C2semidirect
15. #548×C293C2semidirect
129. C2×Dic293#41semidirect
143. Q8×C293#41semidirect
181. C293#3480semidirect
190. #548D293semidirect
307. Q8#17989801semidirect

Maximal subgroups

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

subgroupindex
#340368612
#340368552
#340368502
#340368312
#340368252
#340367982
#340367902
#340366742
#340366652
#340366582
#41×Dic2932
C4×#179898032
C4×#179898012
C2×Q8×Dic2932
#548×C2932
#3480293