Finite Group Databasebeta

← #110715

Subgroups — #110715

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17708C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17327C211yesmaximal, characteristic
5#8078C211yesmaximal, characteristic
6#8076C211yesmaximal, characteristic
7C2×#1299C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1550—22no—
16#1549C412yes—
17#1548C412yes—
18#1542—21no—
19#1542—21no—
20#1531C414yes—
21#1528—22no—
22#1526—21no—
23#1526—21no—
24#1497C414yes—
25#1493—22no—
26#1479—22no—
27#1453—22no—
28#1451—21no—
29#1451—21no—
30C8×M16—21no—
31C8×M16—21no—
32#1301—22no—
33#1301—22no—
34#1299—22no—
35C22×M16—21no—
36C22×M16—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
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#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431C2×C412yes—
68#431D411yescharacteristic
69#431D411yescharacteristic
70#431D412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423C2×C412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81#407—21no—
82#407—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16D412yes—
90C4×M16D411yescharacteristic
91C4×M16D411yescharacteristic
92C2×#97C2×C412yes—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#333—22no—
97#333—22no—
98#332—22no—
99#332—22no—
100#330—24no—
101#330—24no—
102Q8⋊C8—28no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—21no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—21no—
116C2×M16—22no—
117C2×M16—44no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—82no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C4212yes—
132C4⋊Q8C4212yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—22no—
154#117—22no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C414yes—
188C4⋊C8C22⋊C412yes—
189C4⋊C8C22⋊C412yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97C22⋊C412yes—
203#97C22⋊C412yes—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214C24#12611yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4#11711yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4#11711yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245M16—24no—
246M16—44no—
247M16—42no—
248M16—42no—
249M16—42no—
250M16—82no—
251M16—82no—
252M16—82no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C2×C8—82no—
264C2×C8—82no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—42no—
280C42—42no—
281C42C2×C4211yescharacteristic
282C42C2×C22⋊C411yescharacteristic
283C42C2×C22⋊C411yescharacteristic
284C42C2×C22⋊C411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12011yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12611yescharacteristic
296C42#12711yescharacteristic
297C23—22no—
298C23—41no—
299C23—41no—
300C23#38811yescharacteristic
301C23#38811yescharacteristic
302C23#38811yescharacteristic
303Q8—44no—
304Q8—82no—
305Q8—82no—
306Q8—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4#39011yescharacteristic
341C2×C4#39011yescharacteristic
342C2×C4#47112yes—
343C2×C4#47312yes—
344C8—24no—
345C8—42no—
346C8—42no—
347C8—42no—
348C8—44no—
349C8—82no—
350C8—82no—
351C8—82no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—82no—
361C22#143711yescharacteristic
362C22#193012yes—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4#188112yes—
374C4#190812yes—
375C4#193212yes—
376C4#197711yescharacteristic
377C4#197911yescharacteristic
378C2—22no—
379C2—41no—
380C2—41no—
381C2#1094211yescharacteristic
382C2#1094611yescharacteristic
383C2#1095611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#126
LCSG ▷ C42 ▷ 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
#177082
#176482
#174792
#173272
#80782
#80762
C2×#12992