Finite Group Databasebeta

← #108147

Subgroups — #108147

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34085C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17715C211yesmaximal, characteristic
4#8070C211yesmaximal, characteristic
5#7980C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2336C414yescomplemented
13#2332—22no—
14#2331—21no—
15#2327—21no—
16#2323—22no—
17#2298—22no—
18#2295—22no—
19#2295—21no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C42×Q8—21no—
23C4×#117C2211yescharacteristic
24#1995C414yescomplemented
25#1986—22no—
26#1826—22no—
27#1552C414yescomplemented
28#1549—21no—
29#1546—21no—
30#1544—22no—
31#1300—22no—
32#1289—22no—
33#1284—22no—
34#1276—21no—
35#1276—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42#533D412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117D412yes—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8C2×C412yes—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×C4×C8—21no—
76#391—22no—
77#391—24no—
78#389—22no—
79#389—24no—
80#388—24no—
81#388—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#383—21no—
88#383C2×C412yes—
89#382—21no—
90#382—21no—
91#381—21no—
92#381—21no—
93#380—21no—
94#380D412yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4D412yes—
107#376—22no—
108#376—21no—
109#376C2×C412yes—
110#376D412yes—
111C43—22no—
112C43D411yescharacteristic
113C43D411yescharacteristic
114#339—28no—
115#332—24no—
116#332—24no—
117C22×Q8—21no—
118C22×Q8—21no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—22no—
123C2×M16—41no—
124C22×C8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C22⋊C412yes—
130C4⋊Q8C22⋊C412yes—
131C4⋊Q8C4⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133#125—22no—
134#125—22no—
135#125C4214yes—
136#125C4⋊C414yes—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—48no—
154#117—22no—
155#117—22no—
156#117—44no—
157#117C22⋊C414yes—
158#117C4⋊C414yes—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—41no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—22no—
210#97—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—41no—
227#95—41no—
228#95C22⋊C414yes—
229C24#12011yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—44no—
235C2×Q8—82no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
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—44no—
264M16—82no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—82no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—82no—
283C4⋊C4—84no—
284C4⋊C4#9514yes—
285C4⋊C4#9514yes—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C42—24no—
292C42—24no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—44no—
298C42—42no—
299C42—42no—
300C42—48no—
301C42—44no—
302C42—84no—
303C42C2×C4211yescharacteristic
304C42C2×C22⋊C411yescharacteristic
305C42C2×C4⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42#11711yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C23—22no—
320C23—41no—
321C23—41no—
322C23#37911yescharacteristic
323C23#38511yescharacteristic
324C23#38611yescharacteristic
325Q8—44no—
326Q8—44no—
327Q8—82no—
328Q8—82no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C2×#9511yescharacteristic
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#38011yescharacteristic
367C2×C4#38011yescharacteristic
368C2×C4#38111yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38611yescharacteristic
373C2×C4#38611yescharacteristic
374C2×C4#38611yescharacteristic
375C8—82no—
376C8—82no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—82no—
387C22C22⋊#9511yescharacteristic
388C22#179412yes—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—44no—
400C4—44no—
401C4—44no—
402C4—82no—
403C4—84no—
404C4—162no—
405C4#190612yes—
406C4#191212yes—
407C2—22no—
408C2—41no—
409C2—41no—
410C2#1074111yescharacteristic
411C2#1077211yescharacteristic
412C2#1077911yescharacteristic
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
12. #2336C4semidirect
24. #1995C4semidirect
27. #1552C4semidirect

Maximal subgroups

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

subgroupindex
#340852
#218452
#177152
#80702
#79802
#79402
#78692