Finite Group Databasebeta

← #46824896

Subgroups — #46824896

Order 34432.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic538C211yesmaximal, characteristic, complemented
2#31771328C213yesmaximal, complemented
3#31771327C213yesmaximal, complemented
4#31771323C213yesmaximal, complemented
5#31771318C213yesmaximal, complemented
6Q8×C22⋊D269C211yesmaximal, characteristic, complemented
7#31771267C211yesmaximal, characteristic, complemented
8#31771266C211yesmaximal, characteristic, complemented
9#31771227C213yesmaximal, complemented
10#31771226C213yesmaximal, complemented
11#31771159C213yesmaximal, complemented
12#31771133C213yesmaximal, complemented
13#31771081C211yesmaximal, characteristic, complemented
14C4×#16864403C211yesmaximal, characteristic, complemented
15#519×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18#22575453C2213yes—
19#22575452C2213yes—
20#22575452C2213yes—
21#22575451C2213yes—
22#22575448C2216yes—
23#22575448C2213yes—
24#22575447C2216yes—
25#22575446C2213yes—
26#22575443C2213yes—
27#22575442C2213yes—
28#22575441C2213yes—
29#22575441C2216yes—
30#22575439C2216yes—
31#22575439C2211yescharacteristic
32#22575438C2213yes—
33#22575435C2213yes—
34#22575433C2213yes—
35#22575432C2213yes—
36#22575431C2216yes—
37#22575431C2213yes—
38#22575429C2213yes—
39#22575429C2213yes—
40#22575428C2213yes—
41#22575427C2213yes—
42#22575426C2213yes—
43Q8×Dic269C2211yescharacteristic
44Q8×Dic269C2211yescharacteristic
45C4×Dic538C2213yes—
46C4×Dic538C2216yes—
47C4×Dic538C2213yes—
48#22575418C2213yes—
49C4×D1076C2213yes—
50C4×D1076C2213yes—
51#22575415C2213yes—
52C4×C22⋊D269C2213yes—
53C4×C22⋊D269C2213yes—
54#22575411C2213yes—
55#22575411C2213yes—
56#22575410C2216yes—
57#22575410C2213yes—
58C42×D269C2213yes—
59C22×Q8×C269C2211yescharacteristic
60C4×Q8×C269C2213yes—
61C4×Q8×C269C2211yescharacteristic
62C4×Q8×C269C2213yes—
63C4×Q8×C269C2211yescharacteristic
64#117×C269C2213yes—
65#117×C269C2213yes—
66#16864403—24no—
67C2×Dic538C2313yes—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2316yes—
70Q8×D269—24no—
71C2×D1076C2313yes—
72C2×D1076C2313yes—
73C2×C22⋊D269C2311yescharacteristic
74C2×C22⋊D269C2313yes—
75C2×C4×D269C2313yes—
76C2×C4×D269C2313yes—
77C2×C4×D269C2313yes—
78C2×C4×D269C2313yes—
79C4⋊Dic269C2313yes—
80C4⋊Dic269C2313yes—
81C4⋊Dic269C2313yes—
82C4⋊Dic269C2313yes—
83#16864381C2313yes—
84#16864381C2313yes—
85#16864381C2313yes—
86#16864381C2313yes—
87#16864381C2313yes—
88#16864381C2316yes—
89#16864381C2313yes—
90C22⋊Dic269C2311yescharacteristic
91C22⋊Dic269C2313yes—
92#16864379C2311yescharacteristic
93#16864379C2313yes—
94#16864379C2316yes—
95#16864379C2316yes—
96#16864379C2316yes—
97#16864379C2313yes—
98#16864379C2313yes—
99#16864379C2311yescharacteristic
100#16864379C2313yes—
101C4×Dic269C2311yescharacteristic
102C4×Dic269C2313yes—
103C4×Dic269C2313yes—
104C4×Dic269C2316yes—
105C4×Dic269C2313yes—
106C2×Q8×C269—24no—
107C2×Q8×C269C2313yes—
108C2×Q8×C269C2311yescharacteristic
109C22×C4×C269C2313yes—
110C4⋊C4×C269C2313yes—
111C4⋊C4×C269C2313yes—
112C4⋊C4×C269C2313yes—
113C4⋊C4×C269C2313yes—
114C22⋊C4×C269C2313yes—
115C22⋊C4×C269C2311yescharacteristic
116C42×C269C2313yes—
117C42×C269C2316yes—
118C42×C269C2313yes—
119C22×D269C2411yescharacteristic
120C22×D269C2413yes—
121Dic538—26no—
122Dic538—22no—
123Dic538—46no—
124D1076—26no—
125D1076—43no—
126C22⋊D269—22no—
127C22⋊D269—43no—
128C2×Dic269C2411yescharacteristic
129C2×Dic269C2411yescharacteristic
130C2×Dic269C2413yes—
131C2×Dic269C2413yes—
132C2×Dic269C2413yes—
133C2×Dic269C2411yescharacteristic
134C4×D269—26no—
135C4×D269—26no—
136C4×D269—26no—
137C4×D269—43no—
138C23×C269C2411yescharacteristic
139Q8×C269—26no—
140Q8×C269#4114yes—
141C2×C4×C269—26no—
142C2×C4×C269C2413yes—
143C2×C4×C269C2411yescharacteristic
144C2×C4×C269C2411yescharacteristic
145C2×C4×C269C2413yes—
146C2×C4×C269C2413yes—
147C2×C4×C269C2413yes—
148C22×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269C2511yescharacteristic
152C2×D269—21no—
153C2×D269—23no—
154C2×D269—23no—
155C2×D269—21no—
156C2×D269—23no—
157C2×D269—21no—
158C4×C269—23no—
159C4×C269—23no—
160C4×C269—21no—
161C4×C269—21no—
162C4×C269—23no—
163C4×C269C2×#4116yes—
164Dic269—21no—
165Dic269—23no—
166Dic269—21no—
167Dic269—23no—
168Dic269—23no—
169Dic2692−1+412yes—
170C2×C269—41no—
171C2×C269C22×#4111yescharacteristic
172C2×C269C2×2−1+411yescharacteristic
173C2×C269#58511yescharacteristic
174D269—41no—
175D269—43no—
176C269#351811yescharacteristic, complemented, Sylow
177#3518—2691nomaximal, Sylow
178Q82—2691no—
179#556—2693no—
180#556—2693no—
181#556—2693no—
182#555—2693no—
183D4×Q8—2691no—
184#548—2691no—
185#548—2691no—
186#542—2693no—
187#542—2693no—
188#533—2693no—
189#529—2693no—
190#519—2691no—
191#519D26911yescharacteristic, complemented
192C4×#41—2691no—
193C2×#41—2691no—
194C22×Q8—2691no—
195C22×Q8C2×D26911yescharacteristic
196C4⋊Q8—2693no—
197C4⋊Q8—2693no—
198C4⋊Q8—2693no—
199C4⋊Q8—2693no—
200#126—2693no—
201#126—2696no—
202#126—2693no—
203#126—2696no—
204#125—2693no—
205#124—2693no—
206#124—2691no—
207#124—2696no—
208#124—2693no—
209#124—2696no—
210#123—2693no—
211#123—2693no—
212C22⋊Q8—2693no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215C22⋊Q8—2693no—
216C22⋊Q8—2696no—
217C22⋊Q8—2693no—
218#121—2693no—
219#121—2693no—
220#121—2693no—
221C4×Q8—2691no—
222C4×Q8—2691no—
223C4×Q8—2693no—
224C4×Q8—2693no—
225C4×Q8—2693no—
226C4×Q8—2696no—
227C4×Q8C2×D26911yescharacteristic
228C4×Q8C2×D26913yes—
229C4×Q8C2×D26911yescharacteristic
230C4×Q8C2×D26913yes—
231C4×D4—2693no—
232C4×D4—2693no—
233C4×D4—2693no—
234C4×D4—2693no—
235C4×D4—2693no—
236#117—2693no—
237#117—2693no—
238#117—2693no—
239#117—2696no—
240#117C2×D26913yes—
241#117C2×D26913yes—
242C2×C42—2693no—
243#41—5384no—
244C2×Q8—24no—
245C2×Q8—2691no—
246C2×Q8—2693no—
247C2×Q8—2696no—
248C2×Q8—5384no—
249C2×Q8C22×D26913yes—
250C2×Q8C22×D26911yescharacteristic
251C2×D4—2691no—
252C2×D4—2693no—
253C2×D4—2693no—
254C2×D4—2693no—
255C22×C4—2693no—
256C22×C4—2693no—
257C22×C4—2693no—
258C22×C4—2693no—
259C22×C4C22×D26913yes—
260C4⋊C4—2693no—
261C4⋊C4—2693no—
262C4⋊C4—2693no—
263C4⋊C4—2693no—
264C4⋊C4—2693no—
265C4⋊C4—2693no—
266C4⋊C4—2693no—
267C4⋊C4—2693no—
268C4⋊C4—2696no—
269C4⋊C4—2693no—
270C4⋊C4—2693no—
271C4⋊C4C22×D26913yes—
272C4⋊C4C22×D26913yes—
273C4⋊C4C22×D26913yes—
274C4⋊C4C22×D26913yes—
275C22⋊C4—2691no—
276C22⋊C4—2693no—
277C22⋊C4—2693no—
278C22⋊C4—2691no—
279C22⋊C4—2696no—
280C22⋊C4—2696no—
281C22⋊C4—2696no—
282C22⋊C4—2691no—
283C22⋊C4—2693no—
284C22⋊C4—2693no—
285C22⋊C4—2693no—
286C22⋊C4C22×D26913yes—
287C22⋊C4C22×D26911yescharacteristic
288C42—2691no—
289C42—2693no—
290C42—2693no—
291C42—2696no—
292C42—2693no—
293C42C22×D26913yes—
294C42C22×D26916yes—
295C42C22×D26913yes—
296C23—2691no—
297C23—2693no—
298C23C23×D26911yescharacteristic
299Q8—26no—
300Q8—5382no—
301Q8—5386no—
302Q8—10766no—
303Q8#1686440214yes—
304D4—5382no—
305D4—5386no—
306D4—10763no—
307D4—10763no—
308C2×C4—26no—
309C2×C4—2691no—
310C2×C4—2691no—
311C2×C4—2691no—
312C2×C4—2693no—
313C2×C4—2693no—
314C2×C4—2693no—
315C2×C4—5386no—
316C2×C4—5386no—
317C2×C4—5386no—
318C2×C4—10763no—
319C2×C4C23×D26913yes—
320C2×C4C23×D26911yescharacteristic
321C2×C4C23×D26911yescharacteristic
322C2×C4C23×D26913yes—
323C2×C4C23×D26913yes—
324C2×C4C23×D26913yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—5381no—
329C22—5383no—
330C22—5383no—
331C22—5381no—
332C22—5383no—
333C22—5381no—
334C22C24×D26911yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—2692no—
341C4—5381no—
342C4—5383no—
343C4—5381no—
344C4—5383no—
345C4—5383no—
346C4C2×#1686440216yes—
347C2—41no—
348C2—10761no—
349C2—10763no—
350C2C22×#1686440211yescharacteristic
351C22−1+4×D26911yescharacteristic
352C2#3177144211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#519×C269C2
hypercenter#519D269
hypocenterC269#3518
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. Q8×Dic538C2semidirect
2. #31771328C2semidirect
3. #31771327C2semidirect
4. #31771323C2semidirect
5. #31771318C2semidirect
6. Q8×C22⋊D269C2semidirect
7. #31771267C2semidirect
8. #31771266C2semidirect
9. #31771227C2semidirect
10. #31771226C2semidirect
11. #31771159C2semidirect
12. #31771133C2semidirect
13. #31771081C2semidirect
14. C4×#16864403C2semidirect
15. #519×C269C2semidirect
176. C269#3518semidirect
191. #519D269semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5382
#317713282
#317713272
#317713232
#317713182
Q8×C22⋊D2692
#317712672
#317712662
#317712272
#317712262
#317711592
#317711332
#317710812
C4×#168644032
#519×C2692
#3518269