Finite Group Databasebeta

← #108295

Subgroups — #108295

Order 512.

719 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#17579C211yesmaximal, characteristic
3#17567C211yesmaximal, characteristic
4#7965C211yesmaximal, characteristic, complemented
5#7956C211yesmaximal, characteristic, complemented
6#7944C211yesmaximal, characteristic, complemented
7#7894C211yesmaximal, characteristic, complemented
8C2×#529C2211yescharacteristic
9#2994—22no—
10#2993—22no—
11#2983C411yescharacteristic, complemented
12#2982C411yescharacteristic, complemented
13#2975C411yescharacteristic, complemented
14#2974—22no—
15#2972C411yescharacteristic, complemented
16#2969—22no—
17#2966—22no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25#1521—22no—
26#1521—22no—
27#1520—21no—
28#1519—21no—
29#1512C414yescomplemented
30#1508—21no—
31#1507—21no—
32#1500C414yescomplemented
33#1287—22no—
34#1286—22no—
35#1284—22no—
36#1279—22no—
37#529—21no—
38#529—21no—
39#529—21no—
40#529—21no—
41#529C2×C412yes—
42#529C2×C412yes—
43#529D412yes—
44#529D412yes—
45C2×#126—21no—
46C2×#125Q811yescharacteristic
47C2×#124Q811yescharacteristic
48C2×#123—21no—
49C2×C22⋊Q8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4C2×C411yescharacteristic
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#446—22no—
57#446—22no—
58C8×Q8—22no—
59C8×Q8—22no—
60#436—24no—
61#436—22no—
62#436—22no—
63#435—24no—
64#435—24no—
65C8×D4—24no—
66C8×D4—22no—
67C8×D4—22no—
68#433—24no—
69#433—24no—
70#432—21no—
71#432—22no—
72#432—21no—
73#432C2×C412yes—
74#432D412yes—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—22no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431C2×C412yes—
85#431C2×C411yescharacteristic
86#431C2×C411yescharacteristic
87#431C2×C411yescharacteristic
88#431C2×C411yescharacteristic
89#431D412yes—
90#424—24no—
91#423—21no—
92#423—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95#407—22no—
96#407—21no—
97#407D412yes—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101#405—24no—
102C4×M16—21no—
103C4×M16—21no—
104C2×C4×C8—21no—
105C2×C4×C8D412yes—
106#383—21no—
107#383—21no—
108#382—21no—
109#382C2×C412yes—
110#381—21no—
111#381—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119#376C2×C412yes—
120#328—24no—
121#328—24no—
122#327—24no—
123#327—24no—
124C22×Q8C4⋊C411yescharacteristic
125C22×D4—21no—
126C22×D4C4⋊C411yescharacteristic
127C23×C4—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—24no—
132C2×M16—22no—
133C2×M16—41no—
134C22×C8—24no—
135C22×C8—22no—
136C22×C8—41no—
137#126—22no—
138#126—22no—
139#126—22no—
140#126—22no—
141#125—21no—
142#125—21no—
143#125C4⋊C412yes—
144#125C4⋊C412yes—
145#124—21no—
146#124—21no—
147#124C4⋊C412yes—
148#124C4⋊C412yes—
149#123—22no—
150#123—22no—
151#123—22no—
152#123—22no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8C22⋊C412yes—
156C22⋊Q8C22⋊C412yes—
157#121—21no—
158#121—21no—
159#121C22⋊C412yes—
160#121C22⋊C412yes—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8C4212yes—
164C4×Q8C22⋊C412yes—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4C4212yes—
172C4×D4C22⋊C412yes—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C4⋊C411yescharacteristic
184C2×C4⋊C4C4⋊C411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8C22⋊C412yes—
230C22⋊C8C22⋊C412yes—
231#97—24no—
232#97—22no—
233#97—22no—
234C4×C8—24no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—41no—
240C4×C8—41no—
241C4×C8C22⋊C412yes—
242C4×C8C22⋊C412yes—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—21no—
252C24#12111yescharacteristic
253C2×Q8—21no—
254C2×Q8—42no—
255C2×Q8#9512yes—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—82no—
261C2×D4#9512yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—81no—
272C22×C4—82no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4#11711yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291M16—82no—
292M16—84no—
293M16—82no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—84no—
301C2×C8—82no—
302C2×C8—82no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4#9512yes—
316C4⋊C4#9512yes—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C42—22no—
330C42—22no—
331C42—82no—
332C42—82no—
333C42C2×C4211yescharacteristic
334C42C2×C22⋊C411yescharacteristic
335C42C2×C22⋊C411yescharacteristic
336C42C2×C4⋊C411yescharacteristic
337C42C2×C4⋊C411yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12011yescharacteristic
344C42#12011yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42#12311yescharacteristic
348C42#12311yescharacteristic
349C23—22no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—81no—
355C23—81no—
356C23#38011yescharacteristic
357C23#38611yescharacteristic
358C23#38611yescharacteristic
359Q8—42no—
360Q8—42no—
361D4—42no—
362D4—44no—
363D4—42no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4C2×#9511yescharacteristic
398C2×C4C4×C22⋊C411yescharacteristic
399C2×C4C4×C22⋊C411yescharacteristic
400C2×C4#37911yescharacteristic
401C2×C4#38011yescharacteristic
402C2×C4#38111yescharacteristic
403C2×C4#38511yescharacteristic
404C2×C4#38511yescharacteristic
405C2×C4#38511yescharacteristic
406C2×C4#38511yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C8—42no—
410C8—42no—
411C8—42no—
412C8—42no—
413C8—82no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—82no—
427C22—162no—
428C22C22⋊#9511yescharacteristic
429C22#188112yes—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—162no—
445C4—162no—
446C4#181212yes—
447C2—22no—
448C2—41no—
449C2—41no—
450C2—82no—
451C2#1075911yescharacteristic
452C2#1078211yescharacteristic
453C2#1078311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
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
4. #7965C2semidirect
5. #7956C2semidirect
6. #7944C2semidirect
7. #7894C2semidirect
11. #2983C4semidirect
12. #2982C4semidirect
13. #2975C4semidirect
15. #2972C4semidirect
29. #1512C4semidirect
32. #1500C4semidirect

Maximal subgroups

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

subgroupindex
#602902
#175792
#175672
#79652
#79562
#79442
#78942