Finite Group Databasebeta

← #215568

Subgroups — #215568

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9703C211yesmaximal, characteristic, complemented
5#9701C211yesmaximal, characteristic, complemented
6#9697C212yesmaximal, complemented
7#9694C212yesmaximal, complemented
8#9693C211yesmaximal, characteristic, complemented
9#9692C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal
11#8340C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18#2862C2211yescharacteristic
19#2660—21no—
20#2660—21no—
21#2567C414yes—
22#2546—22no—
23#2502C414yes—
24#2434—22no—
25#2317—21no—
26#2317—21no—
27C4×#117C2211yescharacteristic
28#1923—22no—
29#1916—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49C23×Q8—21no—
50#533—42no—
51#533—42no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74#433—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87#398—24no—
88#398—24no—
89#395—24no—
90#395—24no—
91#391—22no—
92#391—22no—
93#391C2×C414yes—
94#391C2×C414yes—
95#389—24no—
96#389—24no—
97#385—24no—
98#385—24no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384C2×C414yes—
104#384C2×C414yes—
105#382—21no—
106#382—21no—
107#382—21no—
108#382—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#333—42no—
124#333—42no—
125#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#332—42no—
130#332—42no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C22×Q8—82no—
139C22×Q8C22×C412yes—
140C23×C4—22no—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—44no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—44no—
160C4⋊Q8—82no—
161C4⋊Q8—82no—
162C4⋊Q8—82no—
163C4⋊Q8—82no—
164#125—44no—
165C22⋊Q8—44no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C4×Q8—44no—
169C4×Q8—42no—
170C4×Q8—42no—
171#117—24no—
172#117—44no—
173#117SD1614yes—
174#117Q1614yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—44no—
222C4⋊C8—44no—
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—
237C24C2×C22⋊C411yescharacteristic
238C2×Q8—24no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—88no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—84no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×SD1612yes—
279C22×C4C2×Q1612yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4#10318yes—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4#10318yes—
317C42—22no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42C2×SD1612yes—
330C42C2×SD1612yes—
331C42C2×Q1612yes—
332C42C2×Q1612yes—
333C42#13714yes—
334C42H(5,𝔽2)12yes—
335C42H(5,𝔽2)12yes—
336C23—21no—
337C23—22no—
338C23—21no—
339C23—22no—
340C23C2×#10312yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—84no—
343Q8—88no—
344Q8—81no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—88no—
384C2×C4C2×#10314yes—
385C2×C4C2×#10314yes—
386C2×C4C2×#10312yes—
387C2×C4#41914yes—
388C2×C4#51812yes—
389C2×C4#53512yes—
390C2×C4#53511yescharacteristic
391C2×C4#53511yescharacteristic
392C2×C4C22×SD1611yescharacteristic
393C2×C4C22×Q1611yescharacteristic
394C2×C4C2×#13712yes—
395C2×C4#57612yes—
396C2×C4#57712yes—
397C8—82no—
398C8—82no—
399C8—82no—
400C8—82no—
401C22—22no—
402C22—24no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22#230911yescharacteristic
407C22#288912yes—
408C22C22×#10311yescharacteristic
409C22C2×#41911yescharacteristic
410C22#289712yes—
411C4—22no—
412C4—22no—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4#320612yes—
424C4#320912yes—
425C4#322011yescharacteristic
426C4#322211yescharacteristic
427C4#322412yes—
428C2—42no—
429C2#1772312yes—
430C2#1776112yes—
431C2#2180211yescharacteristic
432C2#2180411yescharacteristic
433C2#2181311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
4. #9703C2semidirect
5. #9701C2semidirect
6. #9697C2semidirect
7. #9694C2semidirect
8. #9693C2semidirect
9. #9692C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#219222
#208122
#97032
#97012
#96972
#96942
#96932
#96922
#83432
#83402