Finite Group Databasebeta

← #108271

Subgroups — #108271

Order 512.

719 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21831C211yesmaximal, characteristic
3#21425C211yesmaximal, characteristic
4#7996C211yesmaximal, characteristic
5#7965C211yesmaximal, characteristic
6#7888C211yesmaximal, characteristic
7#7862C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983C412yes—
13#2982C412yes—
14#2962—22no—
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#1998C414yes—
24#1997—21no—
25#1997—21no—
26#1997—21no—
27#1966—21no—
28#1901C414yes—
29#1822—22no—
30C4×#101—22no—
31#1291—22no—
32#1287—22no—
33#1278—22no—
34#1275—22no—
35C22×M16C2×C412yes—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8Q812yes—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8D412yes—
51C2×#117D412yes—
52C22×C42C2311yescharacteristic
53#446—24no—
54#446—28no—
55#446—24no—
56#446—24no—
57C8×Q8—24no—
58C8×Q8—24no—
59C8×Q8—24no—
60#433—28no—
61#432—24no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#431—24no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431C2×C411yescharacteristic
80#431C2×C411yescharacteristic
81#431C2×C412yes—
82#431D412yes—
83#431D412yes—
84#424—24no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423C2×C412yes—
89C2×C4⋊C8—21no—
90C2×#101—24no—
91#407—21no—
92#407—21no—
93#407—21no—
94#407C2×C412yes—
95C2×C22⋊C8—21no—
96#405—24no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16C2×C412yes—
104C4×M16D412yes—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8D412yes—
108#343—24no—
109#343—24no—
110#343—24no—
111C22×Q8C22⋊C412yes—
112C22×Q8C4⋊C412yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—44no—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133#125—22no—
134#125—22no—
135#125—22no—
136#125—22no—
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—
153#117—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157C2×C4⋊C4C4214yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C4⋊C412yes—
162C2×C4⋊C4C4⋊C414yes—
163C2×C22⋊C4C4⋊C414yes—
164C2×C42C22⋊C414yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—28no—
178C4⋊C8—28no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190#101—48no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#97—28no—
203#97—22no—
204#97—22no—
205#97—41no—
206#97—41no—
207#97—41no—
208#97—41no—
209C4×C8—28no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8C22⋊C412yes—
217C4×C8C22⋊C412yes—
218C4×C8C22⋊C414yes—
219C24C2×C4⋊C411yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—44no—
225C2×Q8—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4#9514yes—
230C22×C4#9514yes—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249M16—24no—
250M16—24no—
251M16—44no—
252M16—42no—
253M16—42no—
254M16—44no—
255M16—44no—
256M16—44no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C42—24no—
282C42—24no—
283C42C2×C22⋊C411yescharacteristic
284C42C2×C22⋊C411yescharacteristic
285C42#11711yescharacteristic
286C42#11711yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12011yescharacteristic
292C42#12011yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C23—41no—
300C23—41no—
301C23—41no—
302C23C2×#9511yescharacteristic
303C23#38511yescharacteristic
304C23#38511yescharacteristic
305Q8—44no—
306Q8—44no—
307Q8—44no—
308Q8—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4#37911yescharacteristic
334C2×C4#38011yescharacteristic
335C2×C4#38011yescharacteristic
336C2×C4#38111yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38611yescharacteristic
343C8—42no—
344C8—42no—
345C8—44no—
346C8—44no—
347C8—82no—
348C8—82no—
349C8—82no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—82no—
360C22C22⋊#9511yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—44no—
371C4—44no—
372C4#181312yes—
373C4#189112yes—
374C4#191212yes—
375C2—41no—
376C2—41no—
377C2—41no—
378C2#1074111yescharacteristic
379C2#1074611yescharacteristic
380C2#1076011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#602502
#218312
#214252
#79962
#79652
#78882
#78622