Finite Group Databasebeta

← #110699

Subgroups — #110699

Order 512.

555 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17739C211yesmaximal, characteristic
2#17491C211yesmaximal, characteristic
3C2×#1481C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#8002C211yesmaximal, characteristic
7#7965C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#1561—22no—
16#1556C412yes—
17#1556C412yes—
18#1544—22no—
19#1497C414yes—
20#1487—22no—
21#1485—22no—
22#1481—22no—
23#1481C414yes—
24C2×Q8⋊C8—21no—
25C2×Q8⋊C8—21no—
26#1454—22no—
27#1453—22no—
28#1451—21no—
29C8×M16—21no—
30#1301—22no—
31#1292—22no—
32#1287—22no—
33C23×C8—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#432C2×C412yes—
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#431C2×C412yes—
67#431C2×C412yes—
68#431D412yes—
69#431D412yes—
70#431D412yes—
71#423—21no—
72#423—21no—
73#423D412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2×C412yes—
82#407—21no—
83#407—21no—
84#407—21no—
85#407—21no—
86C2×C22⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93#332—24no—
94#332—24no—
95Q8⋊C8—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—82no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4212yes—
123C4⋊Q8C4212yes—
124#125—22no—
125#125—22no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8C22⋊C414yes—
181C4⋊C8C22⋊C414yes—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—41no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—21no—
201C4×C8—41no—
202C4×C8C22⋊C412yes—
203C4×C8C22⋊C412yes—
204C24#12611yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—82no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—81no—
217C22×C4#11711yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235M16—42no—
236M16—42no—
237M16—82no—
238M16—82no—
239M16—82no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—42no—
267C42—42no—
268C42C2×C4211yescharacteristic
269C42C2×C22⋊C411yescharacteristic
270C42C2×C22⋊C411yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12611yescharacteristic
283C42#12711yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23#38811yescharacteristic
288C23#38811yescharacteristic
289C23#38811yescharacteristic
290Q8—44no—
291Q8—44no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4C4×C22⋊C411yescharacteristic
318C2×C4C4×C22⋊C411yescharacteristic
319C2×C4C4×C22⋊C411yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38611yescharacteristic
322C2×C4#38611yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#39011yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#39011yescharacteristic
329C2×C4#47112yes—
330C2×C4#47312yes—
331C8—42no—
332C8—42no—
333C8—44no—
334C8—44no—
335C8—82no—
336C8—82no—
337C8—82no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22#143711yescharacteristic
348C22#193012yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4#188112yes—
361C4#190812yes—
362C4#197711yescharacteristic
363C4#197911yescharacteristic
364C2—22no—
365C2—41no—
366C2—41no—
367C2#1094211yescharacteristic
368C2#1095011yescharacteristic
369C2#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
#177392
#174912
C2×#14812
#173242
#80822
#80022
#79652