Finite Group Databasebeta

← #215571

Subgroups — #215571

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9709C211yesmaximal, characteristic, complemented
5#9707C211yesmaximal, characteristic, complemented
6#9699C211yesmaximal, characteristic, complemented
7#9699C212yesmaximal, complemented
8#9697C211yesmaximal, characteristic, complemented
9#9693C212yesmaximal, complemented
10#8342C212yesmaximal, complemented
11#8341C212yesmaximal
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2862C2211yescharacteristic
15#2862C2211yescharacteristic
16#2858C2211yescharacteristic
17#2858C2211yescharacteristic
18#2640—21no—
19#2640—21no—
20#2563—22no—
21#2547C414yes—
22#2504—22no—
23#2436C414yes—
24#2311—21no—
25#2311—21no—
26C4×#117C2211yescharacteristic
27#1923—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1840—21no—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1818—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39#1750—21no—
40#1746—21no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×Q8C2×C412yes—
51#532—42no—
52#532—42no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#124C2×C414yes—
64C2×#124C2×C414yes—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84#397—24no—
85#397—24no—
86#393—24no—
87#393—24no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391C2×C414yes—
93#391C2×C414yes—
94#390—22no—
95#390—22no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#386—24no—
101#386—24no—
102#382—22no—
103#382—22no—
104#380—24no—
105#380—24no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4D412yes—
110#376D411yescharacteristic
111#376D411yescharacteristic
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#336—22no—
120#336—22no—
121#336—22no—
122#333—42no—
123#333—42no—
124#333—42no—
125#333—42no—
126#332—42no—
127#332—42no—
128#331—42no—
129#331—42no—
130C2×#41—44no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—44no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—44no—
137C22×Q8—84no—
138C22×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—44no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—44no—
163#127—22no—
164#127—22no—
165#124—44no—
166#124—84no—
167#124—84no—
168#121—44no—
169#121—42no—
170#121—42no—
171C4×D4—44no—
172C4×D4—42no—
173C4×D4—42no—
174#117—24no—
175#117—41no—
176#117—41no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—82no—
221C22⋊C8—44no—
222C22⋊C8—82no—
223#97—42no—
224C4×C8—42no—
225C4×C8—82no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24—22no—
235C24C2×C22⋊C411yescharacteristic
236#41—88no—
237#41—88no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—84no—
248C2×Q8—816no—
249C2×D4—24no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—84no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#13514yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—24no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—22no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—42no—
338C42—42no—
339C42H(5,𝔽2)12yes—
340C42H(5,𝔽2)12yes—
341C23—21no—
342C23—22no—
343C23—21no—
344C23—22no—
345C23—42no—
346C23—44no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23#41612yes—
351C23C22×C22⋊C411yescharacteristic
352Q8—84no—
353Q8—84no—
354Q8—88no—
355Q8—82no—
356Q8—82no—
357D4—84no—
358D4—88no—
359D4—82no—
360D4—82no—
361D4—81no—
362D4—81no—
363D4—82no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—88no—
397C2×C4—88no—
398C2×C4#41612yes—
399C2×C4#51812yes—
400C2×C4#53512yes—
401C2×C4#53511yescharacteristic
402C2×C4#53511yescharacteristic
403C2×C4C2×#13512yes—
404C2×C4#57511yescharacteristic
405C2×C4#57511yescharacteristic
406C2×C4#57712yes—
407C2×C4#57812yes—
408C8—82no—
409C8—82no—
410C8—82no—
411C8—82no—
412C22—22no—
413C22—24no—
414C22—41no—
415C22—41no—
416C22—42no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—84no—
421C22—84no—
422C22#230911yescharacteristic
423C22#288812yes—
424C22C2×#41611yescharacteristic
425C22#289611yescharacteristic
426C22#289812yes—
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4#321112yes—
439C4#321112yes—
440C4#322111yescharacteristic
441C4#322311yescharacteristic
442C4#322512yes—
443C2—42no—
444C2—162no—
445C2#1772612yes—
446C2#1776812yes—
447C2#2180111yescharacteristic
448C2#2180611yescharacteristic
449C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21922C2semidirect
3. #20786C2semidirect
4. #9709C2semidirect
5. #9707C2semidirect
6. #9699C2semidirect
7. #9699C2semidirect
8. #9697C2semidirect
9. #9693C2semidirect
10. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#219222
#207862
#97092
#97072
#96992
#96972
#96932
#83422
#83412