Finite Group Databasebeta

← #380993

Subgroups — #380993

Order 512.

1305 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35529C211yesmaximal, characteristic
2#20638C211yesmaximal, characteristic
3#20354C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11348C212yesmaximal, complemented
6#11225C212yesmaximal, complemented
7#11212C212yesmaximal, complemented
8#10749C212yesmaximal
9#10740C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13#2905C2212yes—
14C22×#100C2212yes—
15#2566C412yescomplemented
16#2566C412yescomplemented
17#2563C412yescomplemented
18#2542C412yescomplemented
19#2454—22no—
20#2452—22no—
21#2437—24no—
22#2317C2211yescharacteristic
23#2309C2211yescharacteristic
24#2296C2211yescharacteristic
25C2×#390C2211yescharacteristic
26#2285C2211yescharacteristic
27#1800—22no—
28#1800—22no—
29#1742C2212yes—
30#1740C2212yes—
31#1740C2212yes—
32C2×#342—24no—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C23×D4C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yescomplemented
42C2×#125—21no—
43C2×#125C2×C412yescomplemented
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—22no—
46#518—24no—
47#518—44no—
48#518—44no—
49#518Q818yescomplemented
50C2×C4×D4—44no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×#117C2314yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—24no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D414yes—
64C22×C42C2311yescharacteristic
65#424—44no—
66#423—24no—
67C2×#100—24no—
68C2×#100—24no—
69C2×#100—24no—
70#399—24no—
71#396—24no—
72#396—24no—
73#393—24no—
74#390—22no—
75#390—22no—
76#390C2×C412yes—
77#390C2×C412yes—
78#390C2×C412yes—
79#390C2×C412yes—
80#389—21no—
81#389—21no—
82#389—22no—
83#385—24no—
84#385—24no—
85#385—22no—
86#385—22no—
87#384C2×C411yescharacteristic, complemented
88#384C2×C412yescomplemented
89#384C2×C411yescharacteristic, complemented
90#382C2×C411yescharacteristic, complemented
91#382C2×C412yescomplemented
92#382C2×C411yescharacteristic, complemented
93#380—24no—
94#380—22no—
95#379—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C412yescomplemented
98C4×C4⋊C4C2×C412yescomplemented
99#343—22no—
100#343—22no—
101#342—24no—
102#342—24no—
103#340—44no—
104#339—44no—
105#339—44no—
106H(3,ℤ/4)—44no—
107C25C2×D412yes—
108C22×Q8—22no—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22×C412yes—
113C22×D4C2×D412yes—
114C22×D4C2×Q814yes—
115C22×D4#4114yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—44no—
119C23×C4C22⋊C414yes—
120C23×C4C22⋊C414yes—
121C23×C4C2411yescharacteristic
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C22×C8—24no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#125—44no—
132#125—44no—
133C22⋊Q8—48no—
134C22⋊Q8—88no—
135C4×D4—416no—
136C4×D4—44no—
137C4×D4—44no—
138C4×D4—44no—
139C4×D4—44no—
140C4×D4—416no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—22no—
145#117—22no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117C2×Q818yes—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
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⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—216no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—84no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C22⋊C4C22⋊C414yes—
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C4⋊C8—48no—
196#100—48no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—24no—
202C24—24no—
203C24—22no—
204C24—22no—
205C24—82no—
206C24—82no—
207C24C2×C22⋊C414yes—
208C24C22⋊Q818yes—
209C24C22×D411yescharacteristic
210C2×Q8—44no—
211C2×Q8—84no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—28no—
217C2×D4—28no—
218C2×D4—28no—
219C2×D4—28no—
220C2×D4C4×Q818yes—
221C2×D4#12414yes—
222C2×D4C4⋊Q814yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—416no—
236C22×C4—42no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4—164no—
242C22×C4—164no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×Q812yes—
250C22×C4C2×#4112yes—
251C22×C4H(5,𝔽2)14yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+412yes—
254M16—44no—
255M16—44no—
256M16—44no—
257M16—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—44no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—28no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—28no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—48no—
286C22⋊C4—416no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—84no—
292C22⋊C4—84no—
293C22⋊C4—84no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—24no—
299C42—28no—
300C42—42no—
301C42—42no—
302C42—44no—
303C23—24no—
304C23—24no—
305C23—28no—
306C23—28no—
307C23—41no—
308C23—41no—
309C23—44no—
310C23—44no—
311C23—84no—
312C23—84no—
313C23—88no—
314C23—162no—
315C23—162no—
316C23#39118yes—
317C23C22×C22⋊C411yescharacteristic
318C23C2×C22⋊Q812yes—
319C23#53412yes—
320C23#53511yescharacteristic
321C23#53611yescharacteristic
322Q8—164no—
323D4—216no—
324D4—28no—
325D4—28no—
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—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—48no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—164no—
355C2×C4—164no—
356C2×C4—162no—
357C2×C4—1616no—
358C2×C4C2×C4×Q812yes—
359C2×C4#51812yes—
360C2×C4#52012yes—
361C2×C4C2×#12411yescharacteristic
362C2×C4C2×C4⋊Q811yescharacteristic
363C2×C4#53412yes—
364C2×C4#53712yes—
365C2×C4#53814yes—
366C2×C4#53914yes—
367C2×C4#54314yes—
368C2×C4#54414yes—
369C8—84no—
370C22—21no—
371C22—22no—
372C22—21no—
373C22—22no—
374C22—28no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—48no—
379C22—44no—
380C22—44no—
381C22—88no—
382C22—84no—
383C22—88no—
384C22C2×#39111yescharacteristic
385C22#230811yescharacteristic
386C22#231111yescharacteristic
387C22#262612yes—
388C22#262712yes—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4—88no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4#232914yes—
401C4#263912yes—
402C4#265912yes—
403C2—21no—
404C2—22no—
405C2—21no—
406C2—22no—
407C2—48no—
408C2#1428111yescharacteristic
409C2#2178211yescharacteristic
410C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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. #11348C2semidirect
6. #11225C2semidirect
7. #11212C2semidirect
15. #2566C4semidirect
16. #2566C4semidirect
17. #2563C4semidirect
18. #2542C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
43. C2×#125C2×C4semidirect
49. #518Q8semidirect
87. #384C2×C4semidirect
88. #384C2×C4semidirect
89. #384C2×C4semidirect
90. #382C2×C4semidirect
91. #382C2×C4semidirect
92. #382C2×C4semidirect
97. C4×C4⋊C4C2×C4semidirect
98. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#355292
#206382
#203542
#113882
#113482
#112252
#112122
#107492
#107402