Finite Group Databasebeta

← #107407

Subgroups — #107407

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34239C211yesmaximal, characteristic
2#21314C211yesmaximal, characteristic
3#17667C211yesmaximal, characteristic
4#8097C211yesmaximal, characteristic
5#8048C211yesmaximal, characteristic
6#7960C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2371C414yescomplemented
13#2370—22no—
14#2345—21no—
15#2344—22no—
16#2344—21no—
17#2336—22no—
18#2313—21no—
19C4×#125—22no—
20#2283C2211yescharacteristic
21#2274—21no—
22C4×#117C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1989—22no—
25#1989C414yescomplemented
26#1736—21no—
27#1736—21no—
28#1534—22no—
29#1528—22no—
30#1528C414yescomplemented
31#1303—22no—
32#1297—21no—
33#1297—21no—
34#1286—22no—
35#1284—22no—
36C22×M16—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×#125C2×C412yes—
45C2×#125C2×C412yes—
46C2×#125D412yes—
47C2×#125D412yes—
48C2×#125Q812yes—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73#407—21no—
74#407—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77#389—22no—
78#389—22no—
79#389—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#383—21no—
87#382D412yes—
88#381—22no—
89#381D412yes—
90#380—22no—
91#380—21no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108C4×C22⋊C4D411yescharacteristic
109#376—21no—
110#376C2×C412yes—
111#376C2×C412yes—
112C43—21no—
113C43—21no—
114#340—24no—
115#340—24no—
116#330—28no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—22no—
123C2×M16—44no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C22×C8—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134#125—22no—
135#125—22no—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140#125C4214yes—
141#125C22⋊C414yes—
142#125C4⋊C414yes—
143#125C4⋊C414yes—
144#117—24no—
145#117—44no—
146#117—48no—
147#117—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C4⋊C412yes—
176C2×C4⋊C4C4⋊C412yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—41no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q811yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—22no—
216#97—22no—
217C4×C8—22no—
218C4×C8—41no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—41no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C414yes—
232C24#12011yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267M16—82no—
268M16—82no—
269M16—82no—
270M16—82no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—84no—
296C4⋊C4#9514yes—
297C4⋊C4#9514yes—
298C22⋊C4—24no—
299C22⋊C4—44no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C42—44no—
303C42—48no—
304C42—44no—
305C42—42no—
306C42—42no—
307C42—82no—
308C42—82no—
309C42—81no—
310C42—81no—
311C42—84no—
312C42—82no—
313C42C2×C4211yescharacteristic
314C42C2×C22⋊C411yescharacteristic
315C42C2×C4⋊C411yescharacteristic
316C42C2×C4⋊C411yescharacteristic
317C42#11711yescharacteristic
318C42#11711yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C23—22no—
330C23—22no—
331C23—41no—
332C23#37911yescharacteristic
333C23#38511yescharacteristic
334C23#38611yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4C2×#9511yescharacteristic
376C2×C4C4×C22⋊C411yescharacteristic
377C2×C4C4×C22⋊C411yescharacteristic
378C2×C4#38011yescharacteristic
379C2×C4#38011yescharacteristic
380C2×C4#38111yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38611yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#45412yes—
388C2×C4#45612yes—
389C8—82no—
390C8—82no—
391C8—82no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22C22⋊#9511yescharacteristic
401C22#179312yes—
402C22#190612yes—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—44no—
414C4—44no—
415C4—44no—
416C4—84no—
417C4—82no—
418C4—84no—
419C4#185712yes—
420C4#188611yescharacteristic
421C4#188811yescharacteristic
422C2—22no—
423C2—22no—
424C2—41no—
425C2#1075011yescharacteristic
426C2#1077211yescharacteristic
427C2#1079211yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2371C4semidirect
25. #1989C4semidirect
30. #1528C4semidirect

Maximal subgroups

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

subgroupindex
#342392
#213142
#176672
#80972
#80482
#79602
#79402