Finite Group Databasebeta

← #122142

Subgroups — #122142

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#8102C211yesmaximal, characteristic
3#8049C211yesmaximal, characteristic
4#7982C211yesmaximal, characteristic
5#7982C211yesmaximal, characteristic
6#7946C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10#2842—21no—
11#2840—21no—
12#2665—22no—
13#2665C414yescomplemented
14#2628—22no—
15#2607—22no—
16#2439—22no—
17#2439—22no—
18#2280C2211yescharacteristic
19#2280C2211yescharacteristic
20#2280C2211yescharacteristic
21#2280C2211yescharacteristic
22#2280C2211yescharacteristic
23#1304—21no—
24#1304—21no—
25#1297—22no—
26#1289—22no—
27#1289—22no—
28#1284—22no—
29#1282—22no—
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C23×C8Q812yescomplemented
33C2×C4⋊Q8—22no—
34C2×#117—21no—
35C2×#117—21no—
36C2×#117—21no—
37C2×#117—21no—
38C2×#117—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407Q812yescomplemented
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51#400—24no—
52#399—24no—
53#396—24no—
54#396—24no—
55#395—24no—
56#393—24no—
57#393—24no—
58#393—24no—
59#391—24no—
60#391—24no—
61#389—22no—
62#389—22no—
63#387—24no—
64#387—24no—
65#382—22no—
66#381—21no—
67#381—21no—
68#381C2×C412yes—
69#381D412yes—
70#381Q812yescomplemented
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380C2×C412yes—
79#380D412yes—
80#380D412yes—
81#379—21no—
82#379—21no—
83#379—21no—
84#379—21no—
85#379D411yescharacteristic
86#379D411yescharacteristic
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4D412yes—
95C4×C22⋊C4D412yes—
96C4×C22⋊C4Q812yescomplemented
97C25C2×D411yescharacteristic
98C22×Q8—24no—
99C23×C4—82no—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q811yescharacteristic
107C23×C4C2×Q811yescharacteristic
108C23×C4C2×Q811yescharacteristic
109C23×C4C2×Q811yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—41no—
116C2×M16—82no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C22×C8—82no—
122C4⋊Q8—84no—
123#117—82no—
124#117—82no—
125#117—82no—
126#117—82no—
127#117—82no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C22⋊C412yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—41no—
179#95C22⋊C414yes—
180#95C22⋊C414yes—
181C24—42no—
182C24—81no—
183C24—81no—
184C24C2×C22⋊C411yescharacteristic
185C24#12011yescharacteristic
186C24#12011yescharacteristic
187C24C22⋊Q811yescharacteristic
188C24C22⋊Q811yescharacteristic
189C24#12411yescharacteristic
190C24C4⋊Q811yescharacteristic
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C4×Q811yescharacteristic
217C22×C4C4×Q811yescharacteristic
218C22×C4C4×Q811yescharacteristic
219C22×C4C4×Q811yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4C4⋊Q811yescharacteristic
238C22×C4C4⋊Q811yescharacteristic
239C22×C4C4⋊Q811yescharacteristic
240C22×C4C4⋊Q811yescharacteristic
241C22×C4C4⋊Q811yescharacteristic
242M16—82no—
243M16—82no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—81no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—82no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C42—44no—
277C42—82no—
278C42—82no—
279C42—82no—
280C42—82no—
281C23—22no—
282C23—22no—
283C23—22no—
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—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—88no—
303C23#37911yescharacteristic
304C23#39111yescharacteristic
305C23#39111yescharacteristic
306C23#39311yescharacteristic
307C23#39311yescharacteristic
308C23#39511yescharacteristic
309C23#39511yescharacteristic
310Q8—168no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
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—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#39111yescharacteristic
358C2×C4#39111yescharacteristic
359C2×C4#39111yescharacteristic
360C2×C4#39111yescharacteristic
361C2×C4#39311yescharacteristic
362C2×C4#39311yescharacteristic
363C2×C4#39511yescharacteristic
364C2×C4#39511yescharacteristic
365C8—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—44no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22#144611yescharacteristic
395C22#179412yes—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—82no—
404C4—84no—
405C4—84no—
406C4—82no—
407C4—162no—
408C4—162no—
409C4#203212yes—
410C2—22no—
411C2—41no—
412C2—41no—
413C2—41no—
414C2—41no—
415C2—41no—
416C2—41no—
417C2#1138911yescharacteristic
418C2#1139311yescharacteristic
419C2#1139411yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ 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
13. #2665C4semidirect
32. C23×C8Q8semidirect
42. #407Q8semidirect
70. #381Q8semidirect
96. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#81022
#80492
#79822
#79462
#79272