Finite Group Databasebeta

← #110485

Subgroups — #110485

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21325C211yesmaximal, characteristic
2#17667C211yesmaximal, characteristic
3#17394C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#8109C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7862C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1992—22no—
16#1991—21no—
17#1991—21no—
18#1736C412yes—
19#1736C412yes—
20#1534—22no—
21#1528—22no—
22#1528C414yescomplemented
23#1470—22no—
24#1470C414yescomplemented
25#1463—22no—
26#1455—22no—
27#1453—22no—
28#1452—21no—
29#1450—21no—
30#1305—22no—
31#1285—22no—
32#1275—22no—
33C22×M16C2×C412yes—
34C2×#125—21no—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125C2×C412yes—
45C2×#125C2×C412yes—
46C2×#117C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#407—21no—
73#407—21no—
74#407—21no—
75#407D412yes—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16C2×C412yes—
83C4×M16D412yes—
84C2×#97D412yes—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87#384—21no—
88#382—21no—
89#380—21no—
90#380—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#330—28no—
98#324—28no—
99#322—22no—
100#322—24no—
101C82—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—24no—
112C2×M16—22no—
113C2×M16—42no—
114C2×M16—41no—
115C22×C8C22⋊C412yes—
116C22×C8C22⋊C412yes—
117#125—22no—
118#125—22no—
119#125—22no—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130#125—41no—
131#125—41no—
132#125—41no—
133#117—44no—
134C2×C4⋊C4—21no—
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—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4C4212yes—
160C2×C4⋊C4C4212yes—
161C2×C22⋊C4—22no—
162C2×C42—22no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8C22⋊C414yes—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—41no—
196#97C22⋊C414yes—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205#95—41no—
206#95—41no—
207C24C4×D411yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4C2×C4211yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4#11711yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12711yescharacteristic
239M16—44no—
240M16—44no—
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8#10114yes—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C22⋊C4—82no—
275C42—82no—
276C42C2×C22⋊C411yescharacteristic
277C42#11711yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12011yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12311yescharacteristic
289C42#12311yescharacteristic
290C42#12411yescharacteristic
291C42#12611yescharacteristic
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23C4×C22⋊C411yescharacteristic
297C23#38811yescharacteristic
298C23#39011yescharacteristic
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
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—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4C2×#10112yes—
341C2×C4#41312yes—
342C8—24no—
343C8—44no—
344C8—44no—
345C8—42no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22#143711yescharacteristic
359C22#196611yescharacteristic
360C22#196711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4—162no—
374C4C4×#10112yes—
375C4#190112yes—
376C2—21no—
377C2—21no—
378C2—41no—
379C2—41no—
380C2#1095011yescharacteristic
381C2#1095111yescharacteristic
382C2#1096311yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
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
22. #1528C4semidirect
24. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#213252
#176672
#173942
#173342
#81092
#79532
#78622