Finite Group Databasebeta

← #107231

Subgroups — #107231

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34372C211yesmaximal, characteristic
2#17394C211yesmaximal, characteristic
3C2×#1470C211yesmaximal, characteristic
4#8096C211yesmaximal, characteristic
5#8069C211yesmaximal, characteristic
6#7977C211yesmaximal, characteristic
7#7956C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2382—22no—
14#2371—22no—
15#2345—22no—
16#2345—22no—
17#2336—22no—
18#2336—22no—
19#2313C412yes—
20#2313C412yes—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1470—22no—
24#1470—22no—
25#1470C414yescomplemented
26#1470C414yescomplemented
27#1463—22no—
28C2×#324—22no—
29#1303—22no—
30#1300—22no—
31#1288—22no—
32#1286—22no—
33C2×#125—21no—
34C2×#125—21no—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125C2×C412yes—
42C2×#125C2×C412yes—
43C2×#125D412yes—
44C2×#125Q812yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#432C2×C412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78#389—24no—
79#389—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#383—21no—
89#383D412yes—
90#382—21no—
91#382D412yes—
92#381C2×C412yes—
93#381C2×C412yes—
94#380—22no—
95#380—22no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107#376D412yes—
108#324—28no—
109#324—28no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C2×M16—41no—
116C2×M16—41no—
117C22×C8—41no—
118C22×C8—41no—
119#125—22no—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134#125—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C4212yes—
156C2×C4⋊C4C4212yes—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C4⋊C412yes—
160C2×C4⋊C4C4⋊C412yes—
161C2×C4⋊C4C4⋊C414yes—
162C2×C4⋊C4C4⋊C414yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—41no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—41no—
204C4×C8—41no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95C22⋊C414yes—
216#95C22⋊C414yes—
217C24C2×C22⋊C411yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4#9514yes—
230C22×C4#9514yes—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249M16—82no—
250M16—82no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—82no—
280C42—82no—
281C42—84no—
282C42—84no—
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C23—22no—
300C23—22no—
301C23—41no—
302C23C2×#9511yescharacteristic
303C23#37911yescharacteristic
304C23#38111yescharacteristic
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4C4×C22⋊C411yescharacteristic
339C2×C4#38011yescharacteristic
340C2×C4#38011yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38511yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#38611yescharacteristic
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22C22⋊#9511yescharacteristic
362C22#179412yes—
363C22#183012yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—44no—
378C4—84no—
379C4—84no—
380C2—22no—
381C2—22no—
382C2—41no—
383C2#1074311yescharacteristic
384C2#1075211yescharacteristic
385C2#1077011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
25. #1470C4semidirect
26. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#343722
#173942
C2×#14702
#80962
#80692
#79772
#79562