Finite Group Databasebeta

← #110250

Subgroups — #110250

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20551C211yesmaximal, characteristic
2#17751C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17590C211yesmaximal, characteristic
5#7998C211yesmaximal, characteristic
6#7960C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1771—22no—
16#1768—22no—
17#1753—22no—
18#1561—22no—
19#1552—22no—
20#1549C412yescomplemented
21#1546C412yescomplemented
22#1520—22no—
23#1518C414yescomplemented
24#1515—22no—
25#1511—22no—
26#1506—22no—
27#1506C414yescomplemented
28#1291—22no—
29#1286—22no—
30#1284—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432D412yes—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#431C2×C412yes—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423C2×C412yes—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70#407—21no—
71#407—21no—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76C2×#97—22no—
77C2×C4×C8—22no—
78#383—21no—
79#383—21no—
80#382—21no—
81#382C2×C412yes—
82#381—21no—
83#381D412yes—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90#376C2×C412yes—
91#376D412yes—
92#336—24no—
93#332—24no—
94#332—24no—
95#328—28no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—81no—
107C2×M16—81no—
108C22×C8—22no—
109C22×C8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C4212yes—
117C4⋊Q8C4212yes—
118#125—22no—
119#125—22no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—41no—
141#117—41no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—22no—
154C2×C42—22no—
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×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
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—41no—
176C4⋊C8—41no—
177C4⋊C8C22⋊C414yes—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8C22⋊C414yes—
184#97—22no—
185#97—22no—
186#97—41no—
187#97—41no—
188#97—82no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—82no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—41no—
198#95—41no—
199#95C22⋊C414yes—
200C24#12111yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8—82no—
208C22×C4—24no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233C22×C4#12711yescharacteristic
234M16—82no—
235M16—82no—
236M16—82no—
237M16—82no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—82no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42C2×C4211yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12311yescharacteristic
284C42#12411yescharacteristic
285C42#12611yescharacteristic
286C23—22no—
287C23—41no—
288C23—41no—
289C23#38611yescharacteristic
290C23#38811yescharacteristic
291C23#39011yescharacteristic
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
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—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4C4×C22⋊C411yescharacteristic
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
340C8—44no—
341C8—44no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—82no—
353C22#143711yescharacteristic
354C22#190212yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—84no—
367C4—82no—
368C4—82no—
369C4—162no—
370C4#190411yescharacteristic
371C4#190511yescharacteristic
372C2—22no—
373C2—41no—
374C2—41no—
375C2#1095111yescharacteristic
376C2#1095211yescharacteristic
377C2#1095411yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ 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
20. #1549C4semidirect
21. #1546C4semidirect
23. #1518C4semidirect
27. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#205512
#177512
#175962
#175902
#79982
#79602
#79442