Finite Group Databasebeta

← #391842

Subgroups — #391842

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20640C211yesmaximal, characteristic
4#11382C212yesmaximal, complemented
5#11352C212yesmaximal, complemented
6#11223C212yesmaximal, complemented
7#11206C212yesmaximal, complemented
8#11175C212yesmaximal
9#11173C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#413C2212yes—
14#2879C2212yes—
15#2579C414yes—
16#2452—22no—
17#2451—22no—
18#2440—22no—
19#2439—22no—
20#2409C414yes—
21#2317C2211yescharacteristic
22#2316C2211yescharacteristic
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#1822—22no—
27#1821—22no—
28#1801—22no—
29#1800—22no—
30C2×#344C2214yes—
31C2×#344C2214yes—
32#1742C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2214yes—
36#1728C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2×C412yes—
39C22×M16C2314yes—
40C22×M16C2314yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—44no—
44C2×C22⋊Q8—24no—
45#520—44no—
46C2×C4×Q8D412yes—
47C2×C4×D4C2×C414yes—
48C2×C4×D4D412yes—
49C2×#117D412yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#424—44no—
59#423—24no—
60#413—44no—
61C2×#101—22no—
62C2×#100—22no—
63#408—44no—
64#407—24no—
65#398—24no—
66#396—24no—
67#395—24no—
68#393—24no—
69#391—21no—
70#391—21no—
71#391—24no—
72#390—21no—
73#390—21no—
74#389—22no—
75#389—22no—
76#389—21no—
77#389—21no—
78#389—22no—
79#387—24no—
80#386C2×C414yes—
81#386C2×C412yes—
82#386C2×C412yes—
83#385—24no—
84#385C2×C414yes—
85#385C2×C412yes—
86#385C2×C412yes—
87#384—21no—
88#382—21no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C22⋊C4—22no—
93#344—44no—
94#344—44no—
95#343—24no—
96#343—22no—
97#343—24no—
98#343—22no—
99#343—44no—
100#336—44no—
101C2×#41—24no—
102C2×#41—22no—
103C2×#41—24no—
104C2×#41—22no—
105C22×Q8—24no—
106C22×Q8—44no—
107C22×Q8—84no—
108C22×Q8C2×D411yescharacteristic
109C22×D4C22×C412yes—
110C22×D4C2×D411yescharacteristic
111C23×C4C22⋊C414yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C412yes—
115C23×C4C22×C412yes—
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C22×C8—24no—
131C22×C8—24no—
132C4⋊Q8—48no—
133C4⋊Q8—48no—
134C22⋊Q8—88no—
135C4×Q8—44no—
136C4×D4—44no—
137C4×D4—48no—
138#117—24no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—44no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4C22⋊C414yes—
166C2×C22⋊C4C22×C412yes—
167C2×C22⋊C4C22×C412yes—
168C2×C22⋊C4C22×C412yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—24no—
174C2×C42C22⋊C414yes—
175C2×C42C2×D412yes—
176C4⋊C8—48no—
177#101—44no—
178#100—44no—
179C22⋊C8—48no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—24no—
185#95—24no—
186#95—24no—
187C24C2×C22⋊C412yes—
188C24C23×C411yescharacteristic
189#41—44no—
190#41—44no—
191#41—44no—
192#41—44no—
193C2×Q8—22no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—21no—
197C2×Q8—48no—
198C2×Q8—48no—
199C2×Q8—88no—
200C2×Q8—84no—
201C2×Q8—88no—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—84no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4H(5,𝔽2)14yes—
234C22×C4H(5,𝔽2)12yes—
235C22×C42−1+414yes—
236C22×C42−1+412yes—
237M16—44no—
238M16—44no—
239M16—44no—
240M16—44no—
241M16—44no—
242M16—44no—
243M16—44no—
244M16—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—84no—
273C42—24no—
274C42—44no—
275C42—42no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42—42no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23C22×C22⋊C411yescharacteristic
288C23#51812yes—
289C23#51912yes—
290C23#52012yes—
291Q8—42no—
292Q8—42no—
293Q8—816no—
294D4—48no—
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—48no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—164no—
328C2×C4#53412yes—
329C2×C4#53412yes—
330C2×C4#53511yescharacteristic
331C2×C4#53612yes—
332C2×C4#53611yescharacteristic
333C2×C4#53612yes—
334C2×C4#53712yes—
335C2×C4#56214yes—
336C2×C4#56214yes—
337C2×C4#56314yes—
338C2×C4#56314yes—
339C8—84no—
340C8—84no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—82no—
352C22#230811yescharacteristic
353C22#231011yescharacteristic
354C22#231111yescharacteristic
355C22#237814yes—
356C4—42no—
357C4—44no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—82no—
365C4#281414yes—
366C4#281612yes—
367C4#281712yes—
368C2—21no—
369C2—21no—
370C2—24no—
371C2—82no—
372C2#1466011yescharacteristic
373C2#2177811yescharacteristic
374C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ 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. #11382C2semidirect
5. #11352C2semidirect
6. #11223C2semidirect
7. #11206C2semidirect

Maximal subgroups

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

subgroupindex
#355742
#206952
#206402
#113822
#113522
#112232
#112062
#111752
#111732