Finite Group Databasebeta

← #49134877

Subgroups — #49134877

Order 36224.

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#31827191C211yesmaximal, characteristic, complemented
2#31827184C213yesmaximal, complemented
3#31827155C213yesmaximal, complemented
4#31827149C213yesmaximal, complemented
5Q8×C22⋊D283C211yesmaximal, characteristic, complemented
6#31827122C211yesmaximal, characteristic, complemented
7#31827115C213yesmaximal, complemented
8#31827017C213yesmaximal, complemented
9#31826996C213yesmaximal, complemented
10#31826994C213yesmaximal, complemented
11#41×Dic283C211yesmaximal, characteristic, complemented
12C4×#17985391C213yesmaximal, complemented
13C2×Q8×Dic283C211yesmaximal, characteristic, complemented
14C4×Q8×D283C211yesmaximal, characteristic, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2213yes—
17C2×Q8×D283C2211yescharacteristic
18#22583609C2213yes—
19#22583608C2213yes—
20#22583603C2216yes—
21#22583603C2216yes—
22#22583602C2213yes—
23#22583601C2213yes—
24#22583600C2216yes—
25#22583597C2213yes—
26#22583592C2216yes—
27#22583591C2216yes—
28#22583589C2213yes—
29#22583589C2213yes—
30#22583587C2216yes—
31#22583586C2213yes—
32#22583581C2213yes—
33Q8×Dic283C2213yes—
34Q8×Dic283C2211yescharacteristic
35Q8×Dic283C2211yescharacteristic
36Q8×Dic283C2213yes—
37Q8×Dic283C2211yescharacteristic
38C4×Dic566C2213yes—
39C4×Dic566C2213yes—
40#22583575C2213yes—
41#22583575C2213yes—
42D4×Dic283C2213yes—
43D4×Dic283C2213yes—
44D4×Dic283C2213yes—
45C4×C22⋊D283C2213yes—
46C4×C22⋊D283C2213yes—
47(C22⋊D283)⋊C4C2213yes—
48(C22⋊D283)⋊C4C2211yescharacteristic
49#22583569C2213yes—
50#22583568C2213yes—
51#22583566C2216yes—
52#22583566C2213yes—
53#22583566C2213yes—
54C2×C4⋊Dic283C2213yes—
55C4⋊C4×D283C2213yes—
56C2×C4×Dic283C2213yes—
57C2×C4×Dic283C2213yes—
58C42×D283C2213yes—
59C2×#41×C283C2211yescharacteristic
60C22×Q8×C283C2211yescharacteristic
61#124×C283C2213yes—
62C22⋊Q8×C283C2213yes—
63#121×C283C2213yes—
64C4×Q8×C283C2211yescharacteristic
65C4×D4×C283C2213yes—
66#17985391—212no—
67C2×Dic566C2313yes—
68C2×Dic566C2313yes—
69Q8×D283—24no—
70C2×C22⋊D283C2313yes—
71C2×C22⋊D283C2311yescharacteristic
72C22×Dic283C2311yescharacteristic
73C22×Dic283C2313yes—
74C22×Dic283C2313yes—
75C22×Dic283C2313yes—
76C2×C4×D283C2311yescharacteristic
77C2×C4×D283C2313yes—
78C2×C4×D283C2313yes—
79C4⋊Dic283C2313yes—
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2316yes—
82C4⋊Dic283C2313yes—
83C4⋊Dic283C2313yes—
84Dic283⋊C4C2316yes—
85Dic283⋊C4C2311yescharacteristic
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2313yes—
88Dic283⋊C4C2313yes—
89Dic283⋊C4C2316yes—
90Dic283⋊C4C2316yes—
91C22⋊Dic283C2313yes—
92C22⋊Dic283C2313yes—
93C22⋊Dic283C2316yes—
94C22⋊Dic283C2316yes—
95(C2×D283)⋊C4C2311yescharacteristic
96(C2×D283)⋊C4C2313yes—
97C4×Dic283C2313yes—
98C4×Dic283C2313yes—
99C4×Dic283C2313yes—
100C4×Dic283C2311yescharacteristic
101C4×Dic283C2313yes—
102C4×Dic283C2313yes—
103C4×Dic283C2313yes—
104C4×Dic283C2313yes—
105C4×Dic283C2313yes—
106#41×C283—24no—
107C2×Q8×C283—24no—
108C2×Q8×C283C2313yes—
109C2×Q8×C283C2311yescharacteristic
110C2×D4×C283C2313yes—
111C2×D4×C283C2313yes—
112C22×C4×C283C2313yes—
113C22×C4×C283C2313yes—
114C4⋊C4×C283C2313yes—
115C4⋊C4×C283C2313yes—
116C22⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283C2313yes—
118C22⋊C4×C283C2316yes—
119C42×C283C2313yes—
120C22×D283C2411yescharacteristic
121Dic566—26no—
122Dic566—26no—
123C22⋊D283—26no—
124C22⋊D283#4114yescomplemented
125C2×Dic283—26no—
126C2×Dic283—26no—
127C2×Dic283—22no—
128C2×Dic283—26no—
129C2×Dic283C2411yescharacteristic
130C2×Dic283C2413yes—
131C2×Dic283C2413yes—
132C2×Dic283C2413yes—
133C2×Dic283C2413yes—
134C2×Dic283C2411yescharacteristic
135C2×Dic283C2411yescharacteristic
136C4×D283—26no—
137C4×D283—26no—
138C4×D283—22no—
139C23×C283C2413yes—
140C23×C283C2411yescharacteristic
141Q8×C283—26no—
142Q8×C283#4114yescomplemented
143D4×C283—26no—
144D4×C283—26no—
145C2×C4×C283—26no—
146C2×C4×C283—26no—
147C2×C4×C283C2411yescharacteristic
148C2×C4×C283C2413yes—
149C2×C4×C283C2413yes—
150C2×C4×C283C2411yescharacteristic
151C2×C4×C283C2413yes—
152C22×C283—21no—
153C22×C283—21no—
154C22×C283—23no—
155C22×C283—23no—
156C22×C283—23no—
157C22×C283C2×#4112yes—
158C22×C283C2511yescharacteristic
159C2×D283—21no—
160C2×D283—21no—
161C2×D283C2×#4112yes—
162C4×C283—21no—
163C4×C283—23no—
164C4×C283—23no—
165C4×C283—21no—
166C4×C283C2×#4116yes—
167Dic283—23no—
168Dic283—23no—
169Dic283—23no—
170Dic283—21no—
171Dic283—23no—
172Dic283C2×#4112yes—
173Dic2832−1+412yes—
174C2×C283—22no—
175C2×C283—43no—
176C2×C283C22×#4111yescharacteristic
177C2×C283C22×#4111yescharacteristic
178C2×C283C2×2−1+411yescharacteristic
179D283—22no—
180C283#348211yescharacteristic, complemented, Sylow
181#3482—2831nomaximal, Sylow
182#557—2831no—
183#556—2833no—
184#552—2833no—
185#552—2833no—
186D4×Q8—2831no—
187#548—2831no—
188#548D28311yescharacteristic, complemented
189#547—2833no—
190#533—2833no—
191#529—2833no—
192#529—2833no—
193C4×#41—2831no—
194C4×#41—2833no—
195C2×C4×Q8—2831no—
196C2×C4×Q8—2831no—
197C2×#41—2833no—
198C2×#41C2×D28311yescharacteristic
199C22×Q8—2831no—
200C22×Q8C2×D28311yescharacteristic
201C4⋊Q8—2833no—
202C4⋊Q8—2833no—
203#126—2836no—
204#126—2836no—
205#125—2833no—
206#125—2836no—
207#125—2833no—
208#124—2833no—
209#124C2×D28313yes—
210#123—2836no—
211#123—2836no—
212C22⋊Q8—2833no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8—2836no—
216C22⋊Q8C2×D28313yes—
217#121—2833no—
218#121C2×D28313yes—
219C4×Q8—2831no—
220C4×Q8—2833no—
221C4×Q8—2831no—
222C4×Q8—2833no—
223C4×Q8—2831no—
224C4×Q8—2833no—
225C4×Q8—2833no—
226C4×Q8—2833no—
227C4×Q8—2833no—
228C4×Q8C2×D28311yescharacteristic
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2833no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2831no—
235C4×D4—2833no—
236C4×D4C2×D28313yes—
237#117—2833no—
238#117—2833no—
239#117—2833no—
240#117—2833no—
241#117—2836no—
242C2×C4⋊C4—2833no—
243C2×C4⋊C4—2833no—
244C2×C42—2833no—
245C2×C42—2833no—
246C2×C42—2833no—
247#41—24no—
248#41—56612no—
249C2×Q8—24no—
250C2×Q8—2833no—
251C2×Q8—2833no—
252C2×Q8—5664no—
253C2×Q8C22×D28313yes—
254C2×Q8C22×D28311yescharacteristic
255C2×D4—2831no—
256C2×D4—2833no—
257C2×D4C22×D28313yes—
258C2×D4C22×D28313yes—
259C22×C4—2831no—
260C22×C4—2833no—
261C22×C4—2833no—
262C22×C4—2833no—
263C22×C4—2831no—
264C22×C4—2833no—
265C22×C4—2833no—
266C22×C4C22×D28313yes—
267C22×C4C22×D28313yes—
268C4⋊C4—2836no—
269C4⋊C4—2831no—
270C4⋊C4—2833no—
271C4⋊C4—2833no—
272C4⋊C4—2833no—
273C4⋊C4—2836no—
274C4⋊C4—2836no—
275C4⋊C4—2833no—
276C4⋊C4—2833no—
277C4⋊C4—2836no—
278C4⋊C4—2833no—
279C4⋊C4—2833no—
280C4⋊C4C22×D28313yes—
281C4⋊C4C22×D28313yes—
282C22⋊C4—2833no—
283C22⋊C4—2836no—
284C22⋊C4—2833no—
285C22⋊C4—2836no—
286C22⋊C4—2831no—
287C22⋊C4—2833no—
288C22⋊C4C22×D28311yescharacteristic
289C22⋊C4C22×D28313yes—
290C22⋊C4C22×D28316yes—
291C42—2831no—
292C42—2833no—
293C42—2833no—
294C42—2833no—
295C42—2833no—
296C42—2833no—
297C42—2833no—
298C42—2833no—
299C42—2833no—
300C42C22×D28313yes—
301C23—2831no—
302C23C23×D28313yes—
303C23C23×D28311yescharacteristic
304Q8—26no—
305Q8—5666no—
306Q8—5666no—
307Q8#1798539114yescomplemented
308D4—26no—
309D4—26no—
310D4—2834no—
311D4—5666no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—2831no—
315C2×C4—2831no—
316C2×C4—2831no—
317C2×C4—2833no—
318C2×C4—2833no—
319C2×C4—2833no—
320C2×C4—2833no—
321C2×C4—5662no—
322C2×C4—5666no—
323C2×C4—5666no—
324C2×C4—5662no—
325C2×C4—5666no—
326C2×C4—5666no—
327C2×C4—5666no—
328C2×C4C23×D28311yescharacteristic
329C2×C4C23×D28311yescharacteristic
330C2×C4C23×D28313yes—
331C2×C4C23×D28313yes—
332C2×C4C23×D28313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2832no—
339C22—5661no—
340C22—5661no—
341C22#41×D28312yes—
342C22C24×D28311yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—2832no—
348C4—2832no—
349C4—5663no—
350C4—5663no—
351C4—5663no—
352C4—5661no—
353C4—5663no—
354C4C2×#1798539116yes—
355C2—22no—
356C2—43no—
357C2—5662no—
358C2C2×#41×D28311yescharacteristic
359C2C22×#1798539111yescharacteristic
360C22−1+4×D28311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#548×C283C2
hypercenter#548D283
hypocenterC283#3482
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31827191C2semidirect
2. #31827184C2semidirect
3. #31827155C2semidirect
4. #31827149C2semidirect
5. Q8×C22⋊D283C2semidirect
6. #31827122C2semidirect
7. #31827115C2semidirect
8. #31827017C2semidirect
9. #31826996C2semidirect
10. #31826994C2semidirect
11. #41×Dic283C2semidirect
12. C4×#17985391C2semidirect
13. C2×Q8×Dic283C2semidirect
14. C4×Q8×D283C2semidirect
15. #548×C283C2semidirect
124. C22⋊D283#41semidirect
142. Q8×C283#41semidirect
180. C283#3482semidirect
188. #548D283semidirect
307. Q8#17985391semidirect

Maximal subgroups

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

subgroupindex
#318271912
#318271842
#318271552
#318271492
Q8×C22⋊D2832
#318271222
#318271152
#318270172
#318269962
#318269942
#41×Dic2832
C4×#179853912
C2×Q8×Dic2832
C4×Q8×D2832
#548×C2832
#3482283