Finite Group Databasebeta

← #215575

Subgroups — #215575

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 462 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#9710C211yesmaximal, characteristic, complemented
5#9708C211yesmaximal, characteristic, complemented
6#9700C211yesmaximal, characteristic, complemented
7#9700C212yesmaximal, complemented
8#9698C211yesmaximal, characteristic, complemented
9#9696C212yesmaximal, complemented
10#8341C212yesmaximal
11#8340C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2862C2211yescharacteristic
15#2862C2211yescharacteristic
16#2858C2211yescharacteristic
17#2858C2211yescharacteristic
18#2640—21no—
19#2640—21no—
20#2563C414yes—
21#2547—22no—
22#2504C414yes—
23#2436—22no—
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×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×Q8—21no—
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×#124—22no—
64C2×#124—22no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C412yes—
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#391—22no—
93#391—22no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C414yes—
99#390C2×C414yes—
100#386—24no—
101#386—24no—
102#382C2×C414yes—
103#382C2×C414yes—
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#332—42no—
123#332—42no—
124#332—42no—
125#332—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—21no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×Q8—82no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—44no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—44no—
165#127—22no—
166#127—22no—
167#124—44no—
168#124—82no—
169#124—82no—
170#124—82no—
171#124—82no—
172#121—44no—
173#121—42no—
174#121—42no—
175C4×D4—44no—
176C4×D4—42no—
177C4×D4—42no—
178#117—24no—
179#117—41no—
180#117—41no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—82no—
224C22⋊C8—44no—
225C22⋊C8—82no—
226#97—42no—
227C4×C8—42no—
228C4×C8—82no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—22no—
238C24C2×C22⋊C411yescharacteristic
239#41—88no—
240#41—88no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—88no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—84no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4#13514yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C22⋊C4—84no—
329C22⋊C4—84no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42H(5,𝔽2)12yes—
347C42H(5,𝔽2)12yes—
348C23—21no—
349C23—22no—
350C23—21no—
351C23—22no—
352C23—42no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23#41612yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—88no—
360Q8—82no—
361Q8—82no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—82no—
367Q8—82no—
368D4—84no—
369D4—88no—
370D4—82no—
371D4—82no—
372D4—81no—
373D4—81no—
374D4—82no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—88no—
408C2×C4—88no—
409C2×C4#41612yes—
410C2×C4#51812yes—
411C2×C4#53512yes—
412C2×C4#53511yescharacteristic
413C2×C4#53511yescharacteristic
414C2×C4C2×#13512yes—
415C2×C4#57511yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57612yes—
418C2×C4#57712yes—
419C8—82no—
420C8—82no—
421C8—82no—
422C8—82no—
423C22—22no—
424C22—24no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—84no—
432C22—84no—
433C22#230911yescharacteristic
434C22#288912yes—
435C22C2×#41611yescharacteristic
436C22#289611yescharacteristic
437C22#289712yes—
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4#321112yes—
450C4#321112yes—
451C4#322011yescharacteristic
452C4#322211yescharacteristic
453C4#322412yes—
454C2—42no—
455C2—162no—
456C2#1772712yes—
457C2#1776912yes—
458C2#2180211yescharacteristic
459C2#2180611yescharacteristic
460C2#2181311yescharacteristic
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. #9710C2semidirect
5. #9708C2semidirect
6. #9700C2semidirect
7. #9700C2semidirect
8. #9698C2semidirect
9. #9696C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#219222
#207862
#97102
#97082
#97002
#96982
#96962
#83412
#83402