Finite Group Databasebeta

← #49172401

Subgroups — #49172401

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#34036848C213yesmaximal, complemented
3#34036819C213yesmaximal, complemented
4#34036813C213yesmaximal, complemented
5Q8×C22⋊D293C211yesmaximal, characteristic, complemented
6#34036786C211yesmaximal, characteristic, complemented
7#34036779C213yesmaximal, complemented
8#34036681C213yesmaximal, complemented
9#34036660C213yesmaximal, complemented
10#34036658C213yesmaximal, complemented
11#41×Dic293C211yesmaximal, characteristic, complemented
12C4×#17989801C213yesmaximal, complemented
13C2×Q8×Dic293C211yesmaximal, characteristic, complemented
14C4×Q8×D293C211yesmaximal, characteristic, complemented
15#548×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2213yes—
17C2×Q8×D293C2211yescharacteristic
18#22627623C2213yes—
19#22627622C2213yes—
20#22627617C2216yes—
21#22627617C2216yes—
22#22627616C2213yes—
23#22627615C2213yes—
24#22627614C2216yes—
25#22627611C2213yes—
26#22627606C2216yes—
27#22627605C2216yes—
28#22627603C2213yes—
29#22627603C2213yes—
30#22627601C2216yes—
31#22627600C2213yes—
32#22627595C2213yes—
33Q8×Dic293C2211yescharacteristic
34Q8×Dic293C2211yescharacteristic
35Q8×Dic293C2213yes—
36Q8×Dic293C2211yescharacteristic
37Q8×Dic293C2213yes—
38C4×Dic586C2213yes—
39C4×Dic586C2213yes—
40#22627589C2213yes—
41#22627589C2213yes—
42D4×Dic293C2213yes—
43D4×Dic293C2213yes—
44D4×Dic293C2213yes—
45C4×C22⋊D293C2213yes—
46C4×C22⋊D293C2213yes—
47#22627584C2211yescharacteristic
48#22627584C2213yes—
49#22627583C2213yes—
50#22627582C2213yes—
51#22627580C2213yes—
52#22627580C2213yes—
53#22627580C2216yes—
54C2×C4⋊Dic293C2213yes—
55C4⋊C4×D293C2213yes—
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#17989801—212no—
67C2×Dic586C2313yes—
68C2×Dic586C2313yes—
69Q8×D293—24no—
70C2×C22⋊D293C2313yes—
71C2×C22⋊D293C2311yescharacteristic
72C22×Dic293C2311yescharacteristic
73C22×Dic293C2313yes—
74C22×Dic293C2313yes—
75C22×Dic293C2313yes—
76C2×C4×D293C2311yescharacteristic
77C2×C4×D293C2313yes—
78C2×C4×D293C2313yes—
79C4⋊Dic293C2313yes—
80C4⋊Dic293C2313yes—
81C4⋊Dic293C2313yes—
82C4⋊Dic293C2313yes—
83C4⋊Dic293C2316yes—
84#17989781C2311yescharacteristic
85#17989781C2313yes—
86#17989781C2313yes—
87#17989781C2313yes—
88#17989781C2316yes—
89#17989781C2316yes—
90#17989781C2316yes—
91C22⋊Dic293C2313yes—
92C22⋊Dic293C2313yes—
93C22⋊Dic293C2316yes—
94C22⋊Dic293C2316yes—
95#17989779C2311yescharacteristic
96#17989779C2313yes—
97C4×Dic293C2311yescharacteristic
98C4×Dic293C2313yes—
99C4×Dic293C2313yes—
100C4×Dic293C2313yes—
101C4×Dic293C2313yes—
102C4×Dic293C2313yes—
103C4×Dic293C2313yes—
104C4×Dic293C2313yes—
105C4×Dic293C2313yes—
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×C293C2313yes—
117C22⋊C4×C293C2311yescharacteristic
118C22⋊C4×C293C2316yes—
119C42×C293C2313yes—
120C22×D293C2411yescharacteristic
121Dic586—26no—
122Dic586—26no—
123C22⋊D293—26no—
124C22⋊D293#4114yescomplemented
125C2×Dic293—26no—
126C2×Dic293—26no—
127C2×Dic293—22no—
128C2×Dic293—26no—
129C2×Dic293C2411yescharacteristic
130C2×Dic293C2413yes—
131C2×Dic293C2413yes—
132C2×Dic293C2413yes—
133C2×Dic293C2413yes—
134C2×Dic293C2411yescharacteristic
135C2×Dic293C2411yescharacteristic
136C4×D293—26no—
137C4×D293—26no—
138C4×D293—22no—
139C23×C293C2413yes—
140C23×C293C2411yescharacteristic
141Q8×C293—26no—
142Q8×C293#4114yescomplemented
143D4×C293—26no—
144D4×C293—26no—
145C2×C4×C293—26no—
146C2×C4×C293—26no—
147C2×C4×C293C2411yescharacteristic
148C2×C4×C293C2413yes—
149C2×C4×C293C2413yes—
150C2×C4×C293C2411yescharacteristic
151C2×C4×C293C2413yes—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293—23no—
155C22×C293—23no—
156C22×C293—23no—
157C22×C293C2×#4112yes—
158C22×C293C2511yescharacteristic
159C2×D293—21no—
160C2×D293—21no—
161C2×D293C2×#4112yes—
162C4×C293—21no—
163C4×C293—23no—
164C4×C293—23no—
165C4×C293—21no—
166C4×C293C2×#4116yes—
167Dic293—23no—
168Dic293—23no—
169Dic293—23no—
170Dic293—21no—
171Dic293—23no—
172Dic293C2×#4112yes—
173Dic2932−1+412yes—
174C2×C293—22no—
175C2×C293—43no—
176C2×C293C22×#4111yescharacteristic
177C2×C293C22×#4111yescharacteristic
178C2×C293C2×2−1+411yescharacteristic
179D293—22no—
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#548D29311yescharacteristic, complemented
189#547—2933no—
190#533—2933no—
191#529—2933no—
192#529—2933no—
193C4×#41—2931no—
194C4×#41—2933no—
195C2×C4×Q8—2931no—
196C2×C4×Q8—2931no—
197C2×#41—2933no—
198C2×#41C2×D29311yescharacteristic
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#124C2×D29313yes—
210#123—2936no—
211#123—2936no—
212C22⋊Q8—2933no—
213C22⋊Q8—2933no—
214C22⋊Q8—2933no—
215C22⋊Q8—2936no—
216C22⋊Q8C2×D29313yes—
217#121—2933no—
218#121C2×D29313yes—
219C4×Q8—2931no—
220C4×Q8—2933no—
221C4×Q8—2931no—
222C4×Q8—2933no—
223C4×Q8—2931no—
224C4×Q8—2933no—
225C4×Q8—2933no—
226C4×Q8—2933no—
227C4×Q8—2933no—
228C4×Q8C2×D29311yescharacteristic
229C4×D4—2933no—
230C4×D4—2933no—
231C4×D4—2933no—
232C4×D4—2933no—
233C4×D4—2933no—
234C4×D4—2931no—
235C4×D4—2933no—
236C4×D4C2×D29313yes—
237#117—2933no—
238#117—2933no—
239#117—2933no—
240#117—2933no—
241#117—2936no—
242C2×C4⋊C4—2933no—
243C2×C4⋊C4—2933no—
244C2×C42—2933no—
245C2×C42—2933no—
246C2×C42—2933no—
247#41—24no—
248#41—58612no—
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×D4C22×D29313yes—
258C2×D4C22×D29313yes—
259C22×C4—2931no—
260C22×C4—2933no—
261C22×C4—2933no—
262C22×C4—2933no—
263C22×C4—2931no—
264C22×C4—2933no—
265C22×C4—2933no—
266C22×C4C22×D29313yes—
267C22×C4C22×D29313yes—
268C4⋊C4—2933no—
269C4⋊C4—2933no—
270C4⋊C4—2936no—
271C4⋊C4—2936no—
272C4⋊C4—2931no—
273C4⋊C4—2933no—
274C4⋊C4—2933no—
275C4⋊C4—2933no—
276C4⋊C4—2933no—
277C4⋊C4—2933no—
278C4⋊C4—2936no—
279C4⋊C4—2936no—
280C4⋊C4C22×D29313yes—
281C4⋊C4C22×D29313yes—
282C22⋊C4—2933no—
283C22⋊C4—2936no—
284C22⋊C4—2933no—
285C22⋊C4—2936no—
286C22⋊C4—2931no—
287C22⋊C4—2933no—
288C22⋊C4C22×D29311yescharacteristic
289C22⋊C4C22×D29313yes—
290C22⋊C4C22×D29316yes—
291C42—2931no—
292C42—2933no—
293C42—2933no—
294C42—2933no—
295C42—2933no—
296C42—2933no—
297C42—2933no—
298C42—2933no—
299C42—2933no—
300C42C22×D29313yes—
301C23—2931no—
302C23C23×D29313yes—
303C23C23×D29311yescharacteristic
304Q8—26no—
305Q8—5866no—
306Q8—5866no—
307Q8#1798980114yescomplemented
308D4—26no—
309D4—26no—
310D4—2934no—
311D4—5866no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—2931no—
315C2×C4—2931no—
316C2×C4—2931no—
317C2×C4—2933no—
318C2×C4—2933no—
319C2×C4—2933no—
320C2×C4—2933no—
321C2×C4—5862no—
322C2×C4—5866no—
323C2×C4—5866no—
324C2×C4—5862no—
325C2×C4—5866no—
326C2×C4—5866no—
327C2×C4—5866no—
328C2×C4C23×D29311yescharacteristic
329C2×C4C23×D29311yescharacteristic
330C2×C4C23×D29313yes—
331C2×C4C23×D29313yes—
332C2×C4C23×D29313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2932no—
339C22—5861no—
340C22—5861no—
341C22#41×D29312yes—
342C22C24×D29311yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—2932no—
348C4—2932no—
349C4—5863no—
350C4—5863no—
351C4—5863no—
352C4—5861no—
353C4—5863no—
354C4C2×#1798980116yes—
355C2—22no—
356C2—43no—
357C2—5862no—
358C2C2×#41×D29311yescharacteristic
359C2C22×#1798980111yescharacteristic
360C22−1+4×D29311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D293
commutator, socle, cosocleC22×C293C25
Fitting#548×C293C2
hypercenter#548D293
hypocenterC293#3482
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #34036855C2semidirect
2. #34036848C2semidirect
3. #34036819C2semidirect
4. #34036813C2semidirect
5. Q8×C22⋊D293C2semidirect
6. #34036786C2semidirect
7. #34036779C2semidirect
8. #34036681C2semidirect
9. #34036660C2semidirect
10. #34036658C2semidirect
11. #41×Dic293C2semidirect
12. C4×#17989801C2semidirect
13. C2×Q8×Dic293C2semidirect
14. C4×Q8×D293C2semidirect
15. #548×C293C2semidirect
124. C22⋊D293#41semidirect
142. Q8×C293#41semidirect
180. C293#3482semidirect
188. #548D293semidirect
307. Q8#17989801semidirect

Maximal subgroups

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

subgroupindex
#340368552
#340368482
#340368192
#340368132
Q8×C22⋊D2932
#340367862
#340367792
#340366812
#340366602
#340366582
#41×Dic2932
C4×#179898012
C2×Q8×Dic2932
C4×Q8×D2932
#548×C2932
#3482293