Finite Group Databasebeta

← #49172402

Subgroups — #49172402

Order 37504.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036855C211yesmaximal, characteristic, complemented
2#34036851C213yesmaximal, complemented
3#34036820C213yesmaximal, complemented
4#34036814C213yesmaximal, complemented
5Q8×C22⋊D293C211yesmaximal, characteristic, complemented
6#34036791C211yesmaximal, characteristic, complemented
7#34036790C211yesmaximal, characteristic, complemented
8#34036780C213yesmaximal, complemented
9#34036683C213yesmaximal, complemented
10#34036666C213yesmaximal, complemented
11#34036663C213yesmaximal, complemented
12C4×#17989803C211yesmaximal, characteristic, complemented
13C4×#17989802C213yesmaximal, complemented
14C4×Q8×D293C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989802C2213yes—
18C2×Q8×D293C2211yescharacteristic
19#22627623C2213yes—
20#22627622C2213yes—
21#22627619C2216yes—
22#22627618C2216yes—
23#22627616C2213yes—
24#22627615C2213yes—
25#22627614C2216yes—
26#22627613C2213yes—
27#22627612C2213yes—
28#22627609C2216yes—
29#22627607C2216yes—
30#22627604C2213yes—
31#22627603C2213yes—
32#22627602C2216yes—
33#22627600C2213yes—
34#22627600C2213yes—
35#22627599C2213yes—
36#22627598C2213yes—
37Q8×Dic293C2211yescharacteristic
38Q8×Dic293C2211yescharacteristic
39C4×Dic586C2213yes—
40C4×Dic586C2213yes—
41C4×Dic586C2213yes—
42#22627589C2213yes—
43C4×D1172C2213yes—
44C4×D1172C2213yes—
45C4×D1172C2213yes—
46#22627586C2213yes—
47C4×C22⋊D293C2213yes—
48C4×C22⋊D293C2213yes—
49C4×C22⋊D293C2211yescharacteristic
50C4×C22⋊D293C2213yes—
51#22627583C2213yes—
52#22627582C2213yes—
53#22627581C2213yes—
54#22627581C2213yes—
55#22627581C2216yes—
56C4⋊C4×D293C2213yes—
57C42×D293C2213yes—
58C42×D293C2213yes—
59C22×Q8×C293C2211yescharacteristic
60C4×Q8×C293C2211yescharacteristic
61C4×Q8×C293C2213yes—
62C4×Q8×C293C2211yescharacteristic
63C4×Q8×C293C2213yes—
64C2×C4⋊C4×C293C2213yes—
65C2×C42×C293C2213yes—
66#17989803—24no—
67#17989802—212no—
68C2×Dic586C2313yes—
69C2×Dic586C2313yes—
70Q8×D293—24no—
71C2×D1172C2313yes—
72C2×D1172C2313yes—
73C2×C22⋊D293C2313yes—
74C2×C22⋊D293C2311yescharacteristic
75C2×C4×D293C2311yescharacteristic
76C2×C4×D293C2313yes—
77C2×C4×D293C2313yes—
78C2×C4×D293C2313yes—
79C2×C4×D293C2313yes—
80C2×C4×D293C2313yes—
81C4⋊Dic293C2313yes—
82C4⋊Dic293C2313yes—
83C4⋊Dic293C2313yes—
84C4⋊Dic293C2313yes—
85#17989781C2311yescharacteristic
86#17989781C2313yes—
87#17989781C2313yes—
88#17989781C2313yes—
89#17989781C2316yes—
90#17989781C2316yes—
91#17989781C2316yes—
92C22⋊Dic293C2311yescharacteristic
93C22⋊Dic293C2313yes—
94#17989779C2311yescharacteristic
95#17989779C2313yes—
96#17989779C2313yes—
97#17989779C2313yes—
98#17989779C2316yes—
99#17989779C2316yes—
100#17989779C2316yes—
101C4×Dic293C2313yes—
102C4×Dic293C2313yes—
103C4×Dic293C2311yescharacteristic
104C4×Dic293C2313yes—
105C4×Dic293C2313yes—
106C4×Dic293C2313yes—
107C2×Q8×C293—24no—
108C2×Q8×C293C2313yes—
109C2×Q8×C293C2311yescharacteristic
110C22×C4×C293C2311yescharacteristic
111C22×C4×C293C2313yes—
112C22×C4×C293C2313yes—
113C4⋊C4×C293C2313yes—
114C4⋊C4×C293C2313yes—
115C4⋊C4×C293C2316yes—
116C42×C293C2313yes—
117C42×C293C2313yes—
118C42×C293C2313yes—
119C42×C293C2313yes—
120C22×D293C2413yes—
121C22×D293C2411yescharacteristic
122Dic586—26no—
123Dic586—26no—
124D1172—26no—
125D1172—26no—
126C22⋊D293—26no—
127C22⋊D293#4114yescomplemented
128C2×Dic293C2411yescharacteristic
129C2×Dic293C2411yescharacteristic
130C2×Dic293C2413yes—
131C2×Dic293C2413yes—
132C2×Dic293C2413yes—
133C2×Dic293C2411yescharacteristic
134C4×D293—26no—
135C4×D293—26no—
136C4×D293—26no—
137C4×D293—22no—
138C4×D293—26no—
139C4×D293—26no—
140C23×C293C2411yescharacteristic
141Q8×C293—26no—
142Q8×C293#4114yescomplemented
143C2×C4×C293—26no—
144C2×C4×C293—26no—
145C2×C4×C293—22no—
146C2×C4×C293C2411yescharacteristic
147C2×C4×C293C2413yes—
148C2×C4×C293C2413yes—
149C2×C4×C293C2413yes—
150C2×C4×C293C2413yes—
151C2×C4×C293C2411yescharacteristic
152C22×C293—21no—
153C22×C293—21no—
154C22×C293C2×#4112yes—
155C22×C293C2511yescharacteristic
156C2×D293—21no—
157C2×D293—21no—
158C2×D293—23no—
159C2×D293—23no—
160C2×D293—23no—
161C2×D293C2×#4112yes—
162C4×C293—23no—
163C4×C293—23no—
164C4×C293—23no—
165C4×C293—21no—
166C4×C293C2×#4116yes—
167C4×C2932−1+412yes—
168Dic293—21no—
169Dic293—23no—
170Dic293—23no—
171Dic293—21no—
172Dic293—23no—
173Dic293C2×#4112yes—
174C2×C293—22no—
175C2×C293C22×#4111yescharacteristic
176C2×C293C22×#4111yescharacteristic
177C2×C293C2×2−1+411yescharacteristic
178D293—22no—
179D293—43no—
180C293#348211yescharacteristic, complemented, Sylow
181#3482—2931nomaximal, Sylow
182#557—2931no—
183#556—2933no—
184#552—2933no—
185#552—2933no—
186D4×Q8—2931no—
187#548—2931no—
188#548—2931no—
189#547—2933no—
190#533—2933no—
191#529—2933no—
192#529—2933no—
193C4×#41—2931no—
194C4×#41—2933no—
195C2×C4×Q8—2931no—
196C2×C4×Q8D29311yescharacteristic, complemented
197C2×#41—2931no—
198C2×#41—2933no—
199C22×Q8—2931no—
200C22×Q8C2×D29311yescharacteristic
201C4⋊Q8—2933no—
202C4⋊Q8—2933no—
203#126—2936no—
204#126—2936no—
205#125—2933no—
206#125—2936no—
207#125—2933no—
208#124—2933no—
209#124—2933no—
210#123—2936no—
211#123—2936no—
212C22⋊Q8—2933no—
213C22⋊Q8—2933no—
214C22⋊Q8—2933no—
215C22⋊Q8—2936no—
216C22⋊Q8—2933no—
217#121—2933no—
218#121—2933no—
219C4×Q8—2931no—
220C4×Q8—2931no—
221C4×Q8—2933no—
222C4×Q8—2933no—
223C4×Q8—2933no—
224C4×Q8—2933no—
225C4×Q8C2×D29311yescharacteristic
226C4×Q8C2×D29313yes—
227C4×Q8C2×D29311yescharacteristic
228C4×Q8C2×D29313yes—
229C4×D4—2931no—
230C4×D4—2933no—
231C4×D4—2933no—
232C4×D4—2933no—
233C4×D4—2933no—
234C4×D4—2933no—
235C4×D4—2933no—
236C4×D4—2933no—
237#117—2933no—
238#117—2933no—
239#117—2933no—
240#117—2933no—
241#117—2936no—
242C2×C4⋊C4—2933no—
243C2×C4⋊C4C2×D29313yes—
244C2×C42—2933no—
245C2×C42—2933no—
246C2×C42C2×D29313yes—
247#41—58612no—
248#41—5864no—
249C2×Q8—24no—
250C2×Q8—2933no—
251C2×Q8—2933no—
252C2×Q8—5864no—
253C2×Q8C22×D29313yes—
254C2×Q8C22×D29311yescharacteristic
255C2×D4—2931no—
256C2×D4—2933no—
257C2×D4—2933no—
258C2×D4—2933no—
259C22×C4—2933no—
260C22×C4—2933no—
261C22×C4—2933no—
262C22×C4—2933no—
263C22×C4—2931no—
264C22×C4—2933no—
265C22×C4C22×D29313yes—
266C22×C4C22×D29313yes—
267C22×C4C22×D29311yescharacteristic
268C4⋊C4—2933no—
269C4⋊C4—2936no—
270C4⋊C4—2931no—
271C4⋊C4—2933no—
272C4⋊C4—2933no—
273C4⋊C4—2933no—
274C4⋊C4—2933no—
275C4⋊C4—2933no—
276C4⋊C4—2933no—
277C4⋊C4—2936no—
278C4⋊C4—2936no—
279C4⋊C4C22×D29313yes—
280C4⋊C4C22×D29313yes—
281C4⋊C4C22×D29316yes—
282C22⋊C4—2931no—
283C22⋊C4—2933no—
284C22⋊C4—2931no—
285C22⋊C4—2933no—
286C22⋊C4—2936no—
287C22⋊C4—2933no—
288C22⋊C4—2933no—
289C22⋊C4—2936no—
290C22⋊C4—2936no—
291C42—2933no—
292C42—2933no—
293C42—2933no—
294C42—2933no—
295C42—2931no—
296C42—2933no—
297C42C22×D29313yes—
298C42C22×D29313yes—
299C42C22×D29313yes—
300C42C22×D29313yes—
301C23—2931no—
302C23—2933no—
303C23C23×D29311yescharacteristic
304Q8—26no—
305Q8—5866no—
306Q8—5866no—
307Q8#1798980214yescomplemented
308D4—2934no—
309D4—5866no—
310D4—5866no—
311D4—5866no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—2931no—
316C2×C4—2931no—
317C2×C4—2931no—
318C2×C4—2933no—
319C2×C4—2933no—
320C2×C4—2933no—
321C2×C4—5866no—
322C2×C4—5866no—
323C2×C4—5866no—
324C2×C4—5866no—
325C2×C4—5862no—
326C2×C4—5866no—
327C2×C4C23×D29313yes—
328C2×C4C23×D29311yescharacteristic
329C2×C4C23×D29313yes—
330C2×C4C23×D29313yes—
331C2×C4C23×D29313yes—
332C2×C4C23×D29311yescharacteristic
333C22—21no—
334C22—21no—
335C22—2932no—
336C22—5861no—
337C22—5861no—
338C22—5863no—
339C22—5863no—
340C22—5863no—
341C22#41×D29312yes—
342C22C24×D29311yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—2932no—
348C4—5863no—
349C4—5861no—
350C4—5863no—
351C4—5861no—
352C4—5863no—
353C4C2×#1798980216yes—
354C4#2262767212yes—
355C2—22no—
356C2—5862no—
357C2—11723no—
358C2C2×#41×D29311yescharacteristic
359C2C22×#1798980211yescharacteristic
360C2C2×#2262767211yescharacteristic
subgroupG/H
centerC2×C4C23×D293
commutator, socle, cosocleC22×C293C25
FrattiniC22C24×D293
FittingC2×C4×Q8×C293C2
hypercenterC2×C4×Q8D293
hypocenterC293#3482
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #34036855C2semidirect
2. #34036851C2semidirect
3. #34036820C2semidirect
4. #34036814C2semidirect
5. Q8×C22⋊D293C2semidirect
6. #34036791C2semidirect
7. #34036790C2semidirect
8. #34036780C2semidirect
9. #34036683C2semidirect
10. #34036666C2semidirect
11. #34036663C2semidirect
12. C4×#17989803C2semidirect
13. C4×#17989802C2semidirect
14. C4×Q8×D293C2semidirect
15. C2×C4×Q8×C293C2semidirect
127. C22⋊D293#41semidirect
142. Q8×C293#41semidirect
180. C293#3482semidirect
196. C2×C4×Q8D293semidirect
307. Q8#17989802semidirect

Maximal subgroups

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

subgroupindex
#340368552
#340368512
#340368202
#340368142
Q8×C22⋊D2932
#340367912
#340367902
#340367802
#340366832
#340366662
#340366632
C4×#179898032
C4×#179898022
C4×Q8×D2932
C2×C4×Q8×C2932
#3482293