Finite Group Databasebeta

← #110899

Subgroups — #110899

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20470C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8096C211yesmaximal, characteristic
6#8052C211yesmaximal, characteristic
7#7998C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1767—21no—
16C4×C4⋊C8—21no—
17#1750—22no—
18#1746—21no—
19#1726—21no—
20#1552—21no—
21#1552—21no—
22#1549C412yescomplemented
23#1546C412yescomplemented
24#1544—22no—
25#1520—22no—
26#1515C414yescomplemented
27#1508—22no—
28#1506—22no—
29#1506—21no—
30#1506—21no—
31#1503C414yescomplemented
32#1303—22no—
33#1297—22no—
34#1291—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431C2×C412yes—
68#431D412yes—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C4×M16—21no—
79C4×M16—21no—
80C2×#97—21no—
81C2×#97—22no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84#383C2×C412yes—
85#382—21no—
86#381—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yes—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4D412yes—
94#376—21no—
95#376D411yescharacteristic
96#376D411yescharacteristic
97C43—21no—
98#336—22no—
99#336—22no—
100#332—22no—
101#332—22no—
102#332—22no—
103#332—22no—
104#328—28no—
105#328—24no—
106#328—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C4212yes—
124C4⋊Q8C4212yes—
125#125—22no—
126#125—22no—
127#125—41no—
128#125—41no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145#117—22no—
146#117—22no—
147#117—42no—
148#117—42no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—41no—
188C4⋊C8C22⋊C412yes—
189C4⋊C8C22⋊C412yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C414yes—
198#97—22no—
199#97—22no—
200#97—44no—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—44no—
207C4×C8—41no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—41no—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216C24#12111yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12611yescharacteristic
250C22×C4#12711yescharacteristic
251M16—82no—
252M16—82no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—82no—
281C42—22no—
282C42—22no—
283C42—44no—
284C42—41no—
285C42—41no—
286C42—42no—
287C42—42no—
288C42—82no—
289C42—82no—
290C42C2×C4211yescharacteristic
291C42C2×C22⋊C411yescharacteristic
292C42#11711yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12611yescharacteristic
306C23—22no—
307C23—41no—
308C23—41no—
309C23#38611yescharacteristic
310C23#38811yescharacteristic
311C23#39011yescharacteristic
312Q8—82no—
313Q8—82no—
314Q8—81no—
315Q8—81no—
316Q8—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4C4×C22⋊C411yescharacteristic
352C2×C4C4×C22⋊C411yescharacteristic
353C2×C4C4×C22⋊C411yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#39011yescharacteristic
362C2×C4#39011yescharacteristic
363C8—44no—
364C8—42no—
365C8—42no—
366C8—82no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—82no—
377C22#143711yescharacteristic
378C22#190212yes—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—44no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4—162no—
394C4#190611yescharacteristic
395C4#190611yescharacteristic
396C4#197612yes—
397C2—22no—
398C2—41no—
399C2—41no—
400C2#1094411yescharacteristic
401C2#1094511yescharacteristic
402C2#1095211yescharacteristic
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
22. #1549C4semidirect
23. #1546C4semidirect
26. #1515C4semidirect
31. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#204702
#177152
#175962
#175452
#80962
#80522
#79982