Finite Group Databasebeta

← #109882

Subgroups — #109882

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20534C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#10852C211yesmaximal, characteristic
4#8077C211yesmaximal, characteristic
5#8071C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7949C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1771—21no—
16#1771—21no—
17#1771—21no—
18#1771—21no—
19#1749—21no—
20C4×C22⋊C8—21no—
21C2×#339—21no—
22C2×#339—21no—
23#1573—22no—
24#1571—21no—
25#1570—21no—
26#1569C414yescomplemented
27#1301—21no—
28#1301—21no—
29#1300—22no—
30#1285C414yescomplemented
31#1285C414yescomplemented
32C22×M16—21no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42C2311yescharacteristic
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432—21no—
43#432C2×C412yes—
44#432C2×C412yes—
45#432C2×C412yes—
46#432C2×C412yes—
47#432C2×C412yes—
48#432D412yes—
49#432Q812yes—
50#431—21no—
51#431—21no—
52#431—21no—
53#431—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423C2×C412yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66#407—21no—
67#407—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2×C412yes—
82C4×C4⋊C4C2×C412yes—
83C4×C4⋊C4Q812yes—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86#376—21no—
87C2×#95—21no—
88C43—21no—
89#339—42no—
90#339—42no—
91#336—21no—
92#336—21no—
93#336—21no—
94#336—21no—
95#335—24no—
96#334—24no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—41no—
115#117—22no—
116#117—22no—
117#117—41no—
118#117—41no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—44no—
123C2×C22⋊C4—41no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42C4⋊C412yes—
134C2×C42C4⋊C412yes—
135C2×C42C22×C411yescharacteristic
136C2×C42C22×C411yescharacteristic
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42C2×Q811yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8C22⋊C412yes—
160C4⋊C8C22⋊C412yes—
161C4⋊C8C4⋊C412yes—
162C4⋊C8C4⋊C412yes—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—41no—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—22no—
176#97—81no—
177#97—81no—
178#97C4214yes—
179C4×C8—22no—
180C4×C8—41no—
181C4×C8—81no—
182C4×C8—81no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—41no—
187C24#12311yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4C2×C4⋊C411yescharacteristic
204C22×C4#11711yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×Q811yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4C22⋊Q811yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4#12411yescharacteristic
218C22×C4#12511yescharacteristic
219C22×C4#12611yescharacteristic
220C22×C4#12611yescharacteristic
221M16—42no—
222M16—42no—
223M16—82no—
224M16—82no—
225M16—81no—
226M16—81no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—81no—
243C2×C8—81no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—82no—
259C42—82no—
260C42C2×C4211yescharacteristic
261C42C2×C22⋊C411yescharacteristic
262C42C2×C4⋊C411yescharacteristic
263C42#11711yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×Q811yescharacteristic
268C42C4×Q811yescharacteristic
269C42C4×Q811yescharacteristic
270C42#12111yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42#12511yescharacteristic
275C42C4⋊Q811yescharacteristic
276C23—21no—
277C23—21no—
278C23—22no—
279C23—41no—
280C23#38511yescharacteristic
281C23#38711yescharacteristic
282C23#38811yescharacteristic
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4C4×C22⋊C411yescharacteristic
317C2×C4C4×C4⋊C411yescharacteristic
318C2×C4C4×C4⋊C411yescharacteristic
319C2×C4#38511yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38711yescharacteristic
322C2×C4#38711yescharacteristic
323C2×C4#38711yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#38911yescharacteristic
326C2×C4#38911yescharacteristic
327C2×C4#39111yescharacteristic
328C2×C4#45311yescharacteristic
329C2×C4#45411yescharacteristic
330C2×C4#45511yescharacteristic
331C2×C4#45611yescharacteristic
332C8—44no—
333C8—42no—
334C8—42no—
335C8—42no—
336C8—42no—
337C8—82no—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—42no—
347C22—42no—
348C22#143611yescharacteristic
349C22#185611yescharacteristic
350C22#185711yescharacteristic
351C22#193212yes—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—82no—
364C4—84no—
365C4#185911yescharacteristic
366C4#186011yescharacteristic
367C4#188211yescharacteristic
368C4#188311yescharacteristic
369C2—21no—
370C2—22no—
371C2—21no—
372C2—41no—
373C2#1089011yescharacteristic
374C2#1089711yescharacteristic
375C2#1089911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1436
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #1569C4semidirect
30. #1285C4semidirect
31. #1285C4semidirect

Maximal subgroups

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

subgroupindex
#205342
#178222
#108522
#80772
#80712
#79502
#79492