Finite Group Databasebeta

← #122192

Subgroups — #122192

Order 512.

693 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#8108C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8104C211yesmaximal, characteristic
5#7927C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2842C412yes—
13#2840C412yes—
14#2665—22no—
15#2665—22no—
16#2628—22no—
17#2607—22no—
18#2439—22no—
19#2439—22no—
20#2280C2211yescharacteristic
21#2280C2211yescharacteristic
22#2280C2211yescharacteristic
23#1305—22no—
24#1305—22no—
25#1305—22no—
26#1282—22no—
27#1282—22no—
28#1282—22no—
29C24×C4C2311yescharacteristic
30C22×M16—21no—
31C22×M16—21no—
32C22×M16Q812yescomplemented
33C23×C8—21no—
34C2×C4⋊Q8—22no—
35C2×#117—21no—
36C2×#117—21no—
37C2×#117—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407Q812yescomplemented
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8Q812yescomplemented
61C2×C22⋊C8Q812yescomplemented
62#400—24no—
63#399—24no—
64#396—24no—
65#396—24no—
66#395—24no—
67#393—24no—
68#393—24no—
69#393—24no—
70#391—24no—
71#391—24no—
72#389—22no—
73#389—22no—
74#387—24no—
75#387—24no—
76#382—22no—
77#381—21no—
78#381—21no—
79#381—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380D412yes—
84#380D412yes—
85#380D412yes—
86#379C2×C411yescharacteristic
87#379C2×C412yes—
88#379C2×C412yes—
89#379C2×C411yescharacteristic
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4D412yes—
94C4×C22⋊C4D412yes—
95C4×C22⋊C4D412yes—
96C25C22×C411yescharacteristic
97C22×Q8—24no—
98C23×C4—82no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×Q811yescharacteristic
106C23×C4C2×Q811yescharacteristic
107C23×C4C2×Q811yescharacteristic
108C23×C4C2×Q811yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—82no—
120C2×M16—82no—
121C2×M16—82no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—82no—
129C4⋊Q8—84no—
130#117—82no—
131#117—82no—
132#117—82no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C22⋊C414yes—
149C2×C22⋊C4C22⋊C414yes—
150C2×C22⋊C4C22⋊C412yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180C24—42no—
181C24—81no—
182C24—81no—
183C24C2×C22⋊C411yescharacteristic
184C24C2×C22⋊C411yescharacteristic
185C24C2×C22⋊C411yescharacteristic
186C24C4×Q811yescharacteristic
187C24C4×Q811yescharacteristic
188C24C4×Q811yescharacteristic
189C24C4×Q811yescharacteristic
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—84no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12411yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4C4⋊Q811yescharacteristic
234C22×C4C4⋊Q811yescharacteristic
235C22×C4C4⋊Q811yescharacteristic
236C22×C4C4⋊Q811yescharacteristic
237C22×C4C4⋊Q811yescharacteristic
238C22×C4C4⋊Q811yescharacteristic
239M16—82no—
240M16—82no—
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—82no—
260C4⋊C4—84no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C42—44no—
274C42—82no—
275C42—82no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—88no—
298C23#37911yescharacteristic
299C23#39111yescharacteristic
300C23#39111yescharacteristic
301C23#39111yescharacteristic
302C23#39111yescharacteristic
303C23#39111yescharacteristic
304C23#39111yescharacteristic
305Q8—168no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4#39311yescharacteristic
347C2×C4#39311yescharacteristic
348C2×C4#39311yescharacteristic
349C2×C4#39311yescharacteristic
350C2×C4#39511yescharacteristic
351C2×C4#39511yescharacteristic
352C2×C4#39511yescharacteristic
353C2×C4#39511yescharacteristic
354C8—82no—
355C8—82no—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22#144611yescharacteristic
386C22#179112yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—82no—
395C4—84no—
396C4—84no—
397C4—82no—
398C4#203212yes—
399C2—22no—
400C2—41no—
401C2—41no—
402C2—41no—
403C2—41no—
404C2—41no—
405C2—41no—
406C2#1139111yescharacteristic
407C2#1139411yescharacteristic
408C2#1139511yescharacteristic
subgroupG/H
center, socleC22#1446
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
32. C22×M16Q8semidirect
49. #407Q8semidirect
60. C2×C22⋊C8Q8semidirect
61. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#81082
#81042
#79272
#79252