Finite Group Databasebeta

← #107251

Subgroups — #107251

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21425C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4#8087C211yesmaximal, characteristic
5#8078C211yesmaximal, characteristic
6#7887C211yesmaximal, characteristic
7#7876C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2985—22no—
11#2984—22no—
12#2983C414yes—
13#2962—22no—
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#1998C414yes—
23#1997—21no—
24#1997—21no—
25#1997—21no—
26#1966—21no—
27#1901C414yes—
28C4×#101—22no—
29#1742—22no—
30#1302—22no—
31#1301—22no—
32#1278—22no—
33#1277—22no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16C2×C412yes—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125C2×C412yes—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8Q812yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D412yes—
52C2×#117D412yes—
53C22×C42C2311yescharacteristic
54#446—28no—
55#446—24no—
56#446—28no—
57#446—28no—
58C8×Q8—24no—
59#433—28no—
60#432—24no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67#432C2×C412yes—
68#432D412yes—
69#431—24no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431D412yes—
74#424—24no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423C2×C412yes—
80#423D412yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D412yes—
84C2×#101—24no—
85#407—21no—
86#407C2×C412yes—
87C2×C22⋊C8—21no—
88#405—24no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16—21no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16C2×C412yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#343—24no—
105#343—24no—
106#343—24no—
107C22×Q8C22⋊C412yes—
108C22×Q8C4⋊C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×Q811yescharacteristic
112C2×M16—22no—
113C2×M16—24no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—44no—
120C2×M16—42no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—44no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160C2×C4⋊C4C4214yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C4⋊C412yes—
165C2×C4⋊C4C4⋊C414yes—
166C2×C22⋊C4C4⋊C414yes—
167C2×C42C22⋊C414yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—28no—
181C4⋊C8—28no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8C22⋊C414yes—
191C4⋊C8C22⋊C414yes—
192#101—48no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#97—28no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—41no—
205#97—41no—
206#97—41no—
207#97—41no—
208C4×C8—28no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C24C2×C4⋊C411yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4#9514yes—
227C22×C4#9514yes—
228C22×C4C2×C4211yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246M16—24no—
247M16—24no—
248M16—24no—
249M16—44no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—42no—
254M16—42no—
255M16—44no—
256M16—44no—
257M16—82no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—82no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C42—24no—
285C42—24no—
286C42C2×C22⋊C411yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42#11711yescharacteristic
289C42#11711yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42#12011yescharacteristic
295C42#12011yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C23—22no—
303C23—41no—
304C23—41no—
305C23C2×#9511yescharacteristic
306C23#38511yescharacteristic
307C23#38511yescharacteristic
308Q8—44no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4#37911yescharacteristic
338C2×C4#38011yescharacteristic
339C2×C4#38011yescharacteristic
340C2×C4#38111yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38611yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#45412yes—
348C2×C4#45612yes—
349C8—44no—
350C8—44no—
351C8—44no—
352C8—44no—
353C8—82no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—82no—
364C22C22⋊#9511yescharacteristic
365C22#185712yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—44no—
376C4—44no—
377C4#179312yes—
378C4#188611yescharacteristic
379C4#188811yescharacteristic
380C4#189112yes—
381C2—22no—
382C2—41no—
383C2—41no—
384C2#1074611yescharacteristic
385C2#1075011yescharacteristic
386C2#1076711yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#602512
#214252
#213372
#80872
#80782
#78872
#78762