Finite Group Databasebeta

← #107799

Subgroups — #107799

Order 512.

1125 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#519C211yesmaximal, characteristic
2#20735C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#8092C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14#2276—22no—
15#2276C414yescomplemented
16#2274—22no—
17#2274—21no—
18C42×Q8—22no—
19C42×Q8—22no—
20C4×#117—22no—
21C4×#117C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1823—21no—
24#1805C414yescomplemented
25#1787—21no—
26C4×#103—22no—
27#1491—22no—
28#1487C414yescomplemented
29#1479—22no—
30#1303—22no—
31#1297—22no—
32#1284—22no—
33#1276—22no—
34C23×Q8D411yescharacteristic
35#519—216no—
36C2×C4×Q8—24no—
37C2×C4×Q8—24no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8Q812yes—
50C2×#117—24no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8C2×C412yes—
72C2×#103—24no—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16—21no—
78C2×C4×C8—21no—
79#383—21no—
80#382C2×C412yes—
81#381—21no—
82#380—21no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—28no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—28no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4D412yes—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4D412yes—
99C4×C22⋊C4D412yes—
100#376—21no—
101#376—24no—
102#376—21no—
103#376C2×C412yes—
104C43—24no—
105C43—21no—
106C43D412yes—
107#340—24no—
108Q8⋊C8—28no—
109C22×Q8—28no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C22⋊C412yes—
114C22×Q8C22⋊C412yes—
115C22×Q8C4⋊C412yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—22no—
122C2×M16—41no—
123C22×C8—22no—
124C4×Q8—232no—
125C4×Q8—232no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8C4214yes—
139C4×Q8C22⋊C414yes—
140C4×Q8C4⋊C414yes—
141C4×Q8C4⋊C414yes—
142#117—232no—
143#117—28no—
144#117—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C4⋊C412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—41no—
164C2×C42—28no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—28no—
176C2×C42—22no—
177C2×C42—41no—
178C2×C42C22⋊C414yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×Q811yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197#103—48no—
198C22⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—28no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—41no—
217#95—41no—
218#95C22⋊C414yes—
219C24#12011yescharacteristic
220C2×Q8—28no—
221C2×Q8—28no—
222C2×Q8—28no—
223C2×Q8—28no—
224C2×Q8—28no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8#9514yes—
230C2×Q8#10418yes—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—82no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263M16—82no—
264M16—82no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—44no—
269C4⋊C4—232no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C4⋊C4#9514yes—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—82no—
284C42—232no—
285C42—28no—
286C42—28no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—24no—
291C42—44no—
292C42—44no—
293C42—44no—
294C42—82no—
295C42—82no—
296C42—88no—
297C42C2×C4211yescharacteristic
298C42C2×C22⋊C411yescharacteristic
299C42C2×C4⋊C411yescharacteristic
300C42C2×C4⋊C411yescharacteristic
301C42#11711yescharacteristic
302C42#11711yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C23—22no—
314C23—22no—
315C23—41no—
316C23#37911yescharacteristic
317C23#38511yescharacteristic
318C23#38611yescharacteristic
319Q8—28no—
320Q8—28no—
321Q8—28no—
322Q8—24no—
323Q8#33918yes—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4C2×#9511yescharacteristic
359C2×C4C4×C22⋊C411yescharacteristic
360C2×C4C4×C22⋊C411yescharacteristic
361C2×C4#38011yescharacteristic
362C2×C4#38011yescharacteristic
363C2×C4#38111yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38611yescharacteristic
370C2×C4C2×#10414yes—
371C8—82no—
372C8—82no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22C22⋊#9511yescharacteristic
383C22#179412yes—
384C22#189612yes—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—28no—
393C4—21no—
394C4—21no—
395C4—48no—
396C4—48no—
397C4—44no—
398C4—162no—
399C4C2×#33912yes—
400C4#189612yes—
401C4#190512yes—
402C2—22no—
403C2—22no—
404C2—41no—
405C2#1071911yescharacteristic
406C2#1077911yescharacteristic
407C2#1078611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2276C4semidirect
24. #1805C4semidirect
28. #1487C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5192
#207352
#174492
#80922
#80522
#79402
#78692