Finite Group Databasebeta

← #391760

Subgroups — #391760

Order 512.

1073 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35619C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#20634C211yesmaximal, characteristic
4#11224C212yesmaximal, complemented
5#11212C212yesmaximal, complemented
6#11149C212yesmaximal
7#11074C212yesmaximal, complemented
8#11055C212yesmaximal, complemented
9#10985C212yesmaximal
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#419C2214yes—
14C2×#413C2212yes—
15C2×#410C2212yes—
16#2503C412yescomplemented
17#2502C412yescomplemented
18#2499C412yescomplemented
19#2499C412yescomplemented
20#2447—22no—
21#2446—22no—
22#2419—24no—
23#2317C2211yescharacteristic
24#2316C2212yes—
25#2310C2211yescharacteristic
26#2297C2211yescharacteristic
27C2×#391C2211yescharacteristic
28#2285C2211yescharacteristic
29#1801—22no—
30#1800—22no—
31#1799—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40#536—44no—
41C2×C4⋊Q8—21no—
42C2×#124—21no—
43C2×#123—22no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8D412yes—
48C2×C4×D4D414yes—
49C2×C4×D4D412yes—
50C2×#117C2×C412yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#424—44no—
59#423—24no—
60#419—44no—
61#419—44no—
62C2×#103—44no—
63#413—44no—
64C2×#101—22no—
65C2×#100—22no—
66#410—44no—
67C2×#99—24no—
68#400—24no—
69#398—24no—
70#397—24no—
71#397—24no—
72#391—21no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#390—21no—
80#389—22no—
81#389—44no—
82#387—24no—
83#387—24no—
84#386—22no—
85#385—24no—
86#385—22no—
87#384C2×C412yes—
88#384C2×C411yescharacteristic
89#384C2×C411yescharacteristic
90#382C2×C412yes—
91#382C2×C411yescharacteristic
92#382C2×C411yescharacteristic
93#381—24no—
94#381—22no—
95#380—22no—
96#380—24no—
97C4×C4⋊C4—22no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C2×#95—22no—
101#343—24no—
102#342—24no—
103#339—44no—
104#339—44no—
105H(3,ℤ/4)—44no—
106C2×#41—24no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—22no—
111C22×Q8—21no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8C22×C412yes—
118C22×Q8C2×D411yescharacteristic
119C22×D4C2×D412yes—
120C22×D4C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C22⋊C414yes—
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C22×C8—24no—
134C4⋊Q8—44no—
135#124—44no—
136#123—48no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C4×Q8—44no—
140C4×D4—44no—
141C4×D4—48no—
142#117—24no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C22⋊C414yes—
176C2×C22⋊C4C2×D412yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—24no—
182C2×C42C22⋊C414yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D412yes—
187C4⋊C8—48no—
188#103—416no—
189#101—44no—
190#100—44no—
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—24no—
201#95—44no—
202C24C2×C22⋊C412yes—
203C24C22×D411yescharacteristic
204#41—44no—
205#41—44no—
206#41—44no—
207#41—44no—
208C2×Q8—22no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—22no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4H(5,𝔽2)14yes—
260C22×C4H(5,𝔽2)12yes—
261C22×C4H(5,𝔽2)12yes—
262C22×C42−1+412yes—
263C22×C42−1+412yes—
264M16—44no—
265M16—44no—
266M16—44no—
267M16—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—82no—
286C4⋊C4—84no—
287C4⋊C4—164no—
288C22⋊C4—24no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C42—24no—
299C42—22no—
300C42—22no—
301C42—24no—
302C42—44no—
303C42—42no—
304C42—42no—
305C42—44no—
306C42—44no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23—42no—
314C23C22×C22⋊C411yescharacteristic
315C23#53412yes—
316C23#53511yescharacteristic
317C23#53611yescharacteristic
318C23#53712yes—
319Q8—42no—
320Q8—42no—
321Q8—48no—
322Q8—82no—
323Q8—82no—
324D4—44no—
325D4—44no—
326D4—42no—
327D4—42no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—48no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—164no—
363C2×C4#51812yes—
364C2×C4#52012yes—
365C2×C4#52012yes—
366C2×C4#53412yes—
367C2×C4#53511yescharacteristic
368C2×C4#53611yescharacteristic
369C2×C4#53712yes—
370C2×C4#56014yes—
371C2×C4#56214yes—
372C2×C4#56214yes—
373C2×C4#56314yes—
374C8—84no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22#230811yescharacteristic
388C22#231111yescharacteristic
389C22#231111yescharacteristic
390C22#281312yes—
391C22#281412yes—
392C4—42no—
393C4—42no—
394C4—44no—
395C4—42no—
396C4—42no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4—164no—
403C4#237614yes—
404C4#281312yes—
405C4#281412yes—
406C2—21no—
407C2—21no—
408C2—22no—
409C2—22no—
410C2—82no—
411C2#1465811yescharacteristic
412C2#2178311yescharacteristic
413C2#2178411yescharacteristic
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. #11224C2semidirect
5. #11212C2semidirect
7. #11074C2semidirect
8. #11055C2semidirect
16. #2503C4semidirect
17. #2502C4semidirect
18. #2499C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#356192
#206402
#206342
#112242
#112122
#111492
#110742
#110552
#109852