Finite Group Databasebeta

← #216167

Subgroups — #216167

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#21431C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9647C211yesmaximal, characteristic, complemented
7#9643C211yesmaximal, characteristic, complemented
8#8942C212yesmaximal
9#8941C212yesmaximal
10#8570C212yesmaximal, complemented
11#8568C212yesmaximal, complemented
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2655—22no—
15#2652—22no—
16#2647C411yescharacteristic
17#2645C412yes—
18#2642C411yescharacteristic
19#2629C411yescharacteristic
20#2625C412yes—
21#2621C411yescharacteristic
22#2402—22no—
23#2401—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C4×C22⋊Q8C2212yes—
27#2295C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1924—22no—
31#1917—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125C2×C411yescharacteristic
52C2×C22⋊Q8C2×C412yes—
53C2×C4×Q8C2312yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—42no—
58#423—42no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×#97—21no—
70C2×#97—22no—
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73#401—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#397—22no—
86#397—22no—
87#397—22no—
88#396—22no—
89#391C2×C412yes—
90#391C2×C412yes—
91#391C2×C412yes—
92#391C2×C412yes—
93#389—22no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#380C2×C412yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4D412yes—
115C4×C22⋊C4D412yes—
116#376—22no—
117#376C2×C411yescharacteristic
118#376C2×C411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126Q8⋊C8—42no—
127Q8⋊C8—42no—
128Q8⋊C8—42no—
129Q8⋊C8—42no—
130C22×Q8C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×M16—42no—
135C2×M16—82no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C4⋊Q8—42no—
143#125—42no—
144C22⋊Q8—44no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2411yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—42no—
190C4⋊C8—44no—
191C4⋊C8—82no—
192C4⋊C8—82no—
193C22⋊C8—44no—
194C22⋊C8—42no—
195#97—42no—
196#97—44no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220C24C2×C22⋊C411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—24no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C23×C411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248M16—84no—
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C42—22no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42H(5,𝔽2)12yes—
302C422−1+412yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23C22×C22⋊C411yescharacteristic
309Q8—44no—
310Q8—44no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4#52012yes—
349C2×C4#53511yescharacteristic
350C2×C4#53611yescharacteristic
351C2×C4#53712yes—
352C2×C4#56812yes—
353C2×C4#56812yes—
354C2×C4#57712yes—
355C2×C4#57812yes—
356C8—44no—
357C8—82no—
358C8—81no—
359C8—81no—
360C8—81no—
361C8—81no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22#231111yescharacteristic
368C22#287911yescharacteristic
369C22#287911yescharacteristic
370C22#289812yes—
371C22#290112yes—
372C4—22no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—41no—
379C4—41no—
380C4—82no—
381C4—82no—
382C4—82no—
383C4—82no—
384C4#299514yes—
385C4#322211yescharacteristic
386C4#322311yescharacteristic
387C4#324212yes—
388C2—81no—
389C2#1750612yes—
390C2#1750612yes—
391C2#2185411yescharacteristic
392C2#2185811yescharacteristic
393C2#2187311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9876C2semidirect
5. #9875C2semidirect
6. #9647C2semidirect
7. #9643C2semidirect
10. #8570C2semidirect
11. #8568C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#219782
#214312
#98762
#98752
#96472
#96432
#89422
#89412
#85702
#85682