Finite Group Databasebeta

← #6495512

Subgroups — #6495512

Order 512.

1457 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic
3#33311C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33225C211yesmaximal, characteristic, complemented
7#29845C212yesmaximal
8#29844C212yesmaximal
9#29202C214yesmaximal, complemented
10#28664C214yesmaximal, complemented
11#27900C214yesmaximal
12#25683C214yesmaximal
13#22221C212yesmaximal, complemented
14#22218C212yesmaximal, complemented
15#20462C211yesmaximal, characteristic
16C2×#541C2212yes—
17C2×#499C2214yes—
18C2×#491C2214yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2857C2212yes—
29#2647C2212yes—
30#2642C2212yes—
31#2629C2212yes—
32#2621C2212yes—
33#2507—22no—
34#2506—22no—
35#2506—22no—
36#2497—22no—
37#2496—22no—
38#2496—22no—
39#2446—22no—
40#2429—22no—
41#2330C2212yes—
42#2327C2212yes—
43#2314C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#117C2211yescharacteristic
47#2068C2218yes—
48#2066C2218yes—
49#2036C2218yes—
50#2035C2218yes—
51#1992C2214yes—
52#1990C2212yes—
53#1990C2212yes—
54#1960C2214yes—
55#1956C2214yes—
56#1956C2214yes—
57#1956C2212yes—
58#1956C2212yes—
59#1928C2218yes—
60#1927C2218yes—
61#1865C2218yes—
62#1863C2218yes—
63#1846C2214yes—
64#1845C2212yes—
65#1844C2212yes—
66#1836C2214yes—
67#1768C2212yes—
68#1768C2212yes—
69#1750C2212yes—
70#1750C2212yes—
71#1744C2211yescharacteristic
72#1744C2211yescharacteristic
73C4×#97C2211yescharacteristic
74C42×C8C2211yescharacteristic
75#541—48no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#126—24no—
82C2×#125—24no—
83C2×#125C2312yes—
84C2×#125C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8C2314yes—
87C2×C4×Q8—22no—
88C2×C4×Q8C2314yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#499—48no—
93#491—48no—
94#487—28no—
95#487—28no—
96#431—24no—
97#431—24no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×#103C2318yes—
105C2×#103C2318yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×#97C2312yes—
109C2×#97C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#400—24no—
117#400—24no—
118#400—24no—
119#398—24no—
120#398—24no—
121#398—24no—
122#397—24no—
123#396—24no—
124#391—22no—
125#391—22no—
126#391C2314yes—
127#391C2314yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—24no—
139#389C2314yes—
140#389C2314yes—
141#388—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#382C2312yes—
154#382C2312yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4D412yes—
163C4×C4⋊C4D412yes—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4D412yes—
166C4×C22⋊C4D412yes—
167#376C2311yescharacteristic
168#376C2311yescharacteristic
169C43C2311yescharacteristic
170C43C2311yescharacteristic
171#340C2318yes—
172#340C2318yes—
173#336C2312yes—
174#336C2312yes—
175#336C2312yes—
176#336C2312yes—
177C22×Q8—22no—
178C22×Q8C2414yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8C2412yes—
183C22×C8C2412yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8—84no—
193#126—416no—
194#125—28no—
195#125—28no—
196#125—416no—
197C22⋊Q8—48no—
198C22⋊Q8—84no—
199C4×Q8—48no—
200C4×Q8—84no—
201#117—24no—
202#117—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2414yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245#107—48no—
246#106—48no—
247C4⋊C8—24no—
248C4⋊C8—24no—
249#103—216no—
250#103—216no—
251#103—48no—
252#103—48no—
253C22⋊C8—24no—
254C22⋊C8—24no—
255#97—24no—
256#97—24no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—24no—
260C4×C8—24no—
261C4×C8#4114yes—
262C4×C8#4114yes—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95C2×D412yes—
274#95C2×D412yes—
275#95C2×D412yes—
276#95C2×D412yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D412yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2511yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8C2×#4114yes—
318C2×C8C2×#4114yes—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—48no—
345C22⋊C4—84no—
346C42—22no—
347C42—22no—
348C42—24no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—21no—
353C42—21no—
354C42—22no—
355C42—22no—
356C42—48no—
357C42—84no—
358C42C2×#4112yes—
359C42C2×#4112yes—
360C42H(5,𝔽2)14yes—
361C422−1+414yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23C23×D411yescharacteristic
368Q8—48no—
369Q8—48no—
370Q8—88no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4#53212yes—
405C2×C4#53212yes—
406C2×C4#53514yes—
407C2×C4#53614yes—
408C2×C4#57714yes—
409C2×C4#57814yes—
410C2×C4C22×#4111yescharacteristic
411C2×C4C22×#4111yescharacteristic
412C2×C4C2×H(5,𝔽2)12yes—
413C2×C4C2×2−1+412yes—
414C8—24no—
415C8—24no—
416C8—42no—
417C8—42no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22C2×#53211yescharacteristic
424C22C2×#53511yescharacteristic
425C22C2×#53611yescharacteristic
426C22C2×#57712yes—
427C22C2×#57812yes—
428C4—24no—
429C4—24no—
430C4—41no—
431C4—41no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—84no—
436C4—84no—
437C4—84no—
438C4—84no—
439C4#317414yes—
440C4#317714yes—
441C4#351112yes—
442C4#351112yes—
443C2—81no—
444C2#3529811yescharacteristic
445C2C2×#317411yescharacteristic
446C2C2×#317711yescharacteristic
447C2#6082412yes—
448C2#6087812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33311C2semidirect
4. #33294C2semidirect
5. #33232C2semidirect
6. #33225C2semidirect
9. #29202C2semidirect
10. #28664C2semidirect
13. #22221C2semidirect
14. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#352842
#333112
#332942
#332322
#332252
#298452
#298442
#292022
#286642
#279002
#256832
#222212
#222182
#204622