Finite Group Databasebeta

← #380996

Subgroups — #380996

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#20358C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11347C212yesmaximal, complemented
6#11224C212yesmaximal, complemented
7#11211C212yesmaximal, complemented
8#10751C212yesmaximal
9#10742C212yesmaximal
10C2×#568C2212yes—
11C2×#544C2211yescharacteristic
12C2×#520C2211yescharacteristic
13C2×#410C2212yes—
14#2568C412yescomplemented
15#2566C412yescomplemented
16#2565C412yescomplemented
17#2563C412yescomplemented
18#2450—24no—
19#2442—22no—
20#2439—22no—
21#2317C2211yescharacteristic
22#2316C2212yes—
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#1799—24no—
28#1742—22no—
29#1742C2212yes—
30C2×#343—22no—
31C2×#343C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38#568—24no—
39#568—44no—
40#568Q818yescomplemented
41C2×#131—44no—
42C22×M16—24no—
43C22×M16C2314yes—
44#544—44no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×#125—21no—
48C2×#125C2×C412yes—
49C2×C22⋊Q8—22no—
50#520—44no—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D414yes—
54C2×#117D412yes—
55C2×#117C2314yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#410—24no—
64#410—24no—
65C2×#99—24no—
66#400—24no—
67#398—24no—
68#396—24no—
69#393—24no—
70#391—24no—
71#391C2×C411yescharacteristic
72#391C2×C412yes—
73#391C2×C411yescharacteristic
74#390C2×C411yescharacteristic
75#390C2×C412yes—
76#390C2×C411yescharacteristic
77#389—22no—
78#389—21no—
79#389—21no—
80#389—44no—
81#387—24no—
82#386—22no—
83#386—22no—
84#385—24no—
85#385—22no—
86#385—22no—
87#384C2×C412yes—
88#382—22no—
89#382C2×C412yes—
90#382C2×C412yes—
91#382C2×C412yes—
92#380—24no—
93#380—22no—
94#380—24no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C22⋊C4—22no—
98C2×#95—22no—
99#343—22no—
100#343—24no—
101#343—22no—
102#342—24no—
103#340—44no—
104#339—44no—
105#339—44no—
106H(3,ℤ/4)—44no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41C2×Q814yes—
111C2×#41#4114yes—
112C22×Q8—22no—
113C22×Q8C22×C411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C22×D4C2×D412yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C22⋊C414yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
122#131—416no—
123#131—416no—
124C2×M16—24no—
125C2×M16—216no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—44no—
131C2×M16—48no—
132C2×M16C2×Q818yes—
133C22×C8—24no—
134C22×C8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137#125—44no—
138#125—44no—
139C22⋊Q8—48no—
140C4×Q8—44no—
141C4×D4—44no—
142C4×D4—48no—
143#117—24no—
144#117—22no—
145#117—22no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—24no—
151#117—44no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42C22⋊C414yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191#99—48no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—44no—
201C24C2×C22⋊C412yes—
202C24C22×D411yescharacteristic
203#41—28no—
204#41—28no—
205#41—28no—
206#41—28no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—44no—
210C2×Q8—84no—
211C2×Q8C4×Q814yes—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4C4×Q814yes—
217C2×D4#12414yes—
218C2×D4C4⋊Q814yes—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—84no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C22⋊Q818yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×Q812yes—
246C22×C4C2×#4112yes—
247C22×C4H(5,𝔽2)14yes—
248C22×C4H(5,𝔽2)12yes—
249C22×C42−1+412yes—
250M16—28no—
251M16—28no—
252M16—416no—
253M16—44no—
254M16—44no—
255C2×C8—28no—
256C2×C8—416no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—164no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—84no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C42—24no—
289C42—22no—
290C42—22no—
291C42—24no—
292C42—44no—
293C42—42no—
294C42—41no—
295C42—41no—
296C42—44no—
297C42—44no—
298C23—22no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23#39114yes—
305C23C22×C22⋊C411yescharacteristic
306C23C2×#12411yescharacteristic
307C23C2×C4⋊Q811yescharacteristic
308C23#53412yes—
309C23#53712yes—
310Q8—28no—
311Q8—164no—
312D4—28no—
313D4—28no—
314D4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—164no—
346C2×C4—162no—
347C2×C4#39114yes—
348C2×C4C2×C4×Q812yes—
349C2×C4#51812yes—
350C2×C4#52012yes—
351C2×C4C2×C22⋊Q812yes—
352C2×C4#53412yes—
353C2×C4#53511yescharacteristic
354C2×C4#53611yescharacteristic
355C2×C4#53814yes—
356C2×C4#53914yes—
357C2×C4#54314yes—
358C2×C4#54414yes—
359C8—48no—
360C22—21no—
361C22—22no—
362C22—21no—
363C22—22no—
364C22—24no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—82no—
372C22C2×#39111yescharacteristic
373C22#230811yescharacteristic
374C22#231111yescharacteristic
375C22#263912yes—
376C22#265912yes—
377C4—24no—
378C4—44no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4—164no—
388C4#232914yes—
389C4#262612yes—
390C4#262712yes—
391C2—21no—
392C2—22no—
393C2—21no—
394C2—22no—
395C2—44no—
396C2#1428111yescharacteristic
397C2#2178411yescharacteristic
398C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11388C2semidirect
5. #11347C2semidirect
6. #11224C2semidirect
7. #11211C2semidirect
14. #2568C4semidirect
15. #2566C4semidirect
16. #2565C4semidirect
17. #2563C4semidirect
40. #568Q8semidirect

Maximal subgroups

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

subgroupindex
#356202
#206342
#203582
#113882
#113472
#112242
#112112
#107512
#107422