Finite Group Databasebeta

← #122173

Subgroups — #122173

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34644C211yesmaximal, characteristic
2#8113C211yesmaximal, characteristic
3#8092C211yesmaximal, characteristic
4#8071C211yesmaximal, characteristic
5#8069C211yesmaximal, characteristic
6#7956C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2667—21no—
13#2667C412yes—
14#2666—22no—
15#2666—22no—
16#2665—21no—
17#2665C412yes—
18#2402—21no—
19#2399—21no—
20#2396—22no—
21#2295—22no—
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1306—21no—
26#1306—21no—
27#1303—22no—
28#1300—22no—
29#1300—21no—
30#1300—21no—
31#1286—22no—
32#1282—22no—
33C2×C4×Q8—22no—
34C2×#117—21no—
35C22×C42C2311yescharacteristic
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432Q812yescomplemented
43#432Q812yescomplemented
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431—21no—
51#431Q812yescomplemented
52#423—21no—
53#423—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8Q812yescomplemented
56#407—21no—
57#407—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C4×M16—21no—
61C4×M16—21no—
62C2×#97—21no—
63C2×C4×C8—21no—
64#400—22no—
65#400—22no—
66#400—22no—
67#399—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#396—22no—
75#396—22no—
76#395—22no—
77#395—22no—
78#395—22no—
79#393—22no—
80#391—22no—
81#391—22no—
82#389—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#383—21no—
89#383—21no—
90#382—21no—
91#382D412yes—
92#381C2×C411yescharacteristic
93#381C2×C411yescharacteristic
94#381D411yescharacteristic
95#381D411yescharacteristic
96#380—21no—
97#380C2×C412yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4C2×C412yes—
108C4×C22⋊C4D412yes—
109#376—21no—
110#376D411yescharacteristic
111#376D411yescharacteristic
112#376D412yes—
113C43—21no—
114C22×Q8—22no—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—41no—
119C2×M16—41no—
120C22×C8—22no—
121C22×C8—41no—
122C4×Q8—48no—
123#117—44no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4C22⋊C412yes—
140C2×C22⋊C4C22⋊C412yes—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42C2×Q811yescharacteristic
153C2×C42C2×Q811yescharacteristic
154C2×C42C2×Q811yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175#97—22no—
176#97—22no—
177#97—41no—
178#97—41no—
179C4×C8—22no—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—41no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—21no—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205C24C2×C22⋊C411yescharacteristic
206C2×Q8—44no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C4×Q811yescharacteristic
221C22×C4C4×Q811yescharacteristic
222C22×C4C4×Q811yescharacteristic
223C22×C4C4×Q811yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4C4⋊Q811yescharacteristic
235C22×C4C4⋊Q811yescharacteristic
236M16—82no—
237M16—82no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C42—24no—
262C42—42no—
263C42—44no—
264C42—44no—
265C42—82no—
266C42—82no—
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42#12411yescharacteristic
276C42#12411yescharacteristic
277C42#12411yescharacteristic
278C42#12411yescharacteristic
279C42C4⋊Q811yescharacteristic
280C42C4⋊Q811yescharacteristic
281C42C4⋊Q811yescharacteristic
282C42C4⋊Q811yescharacteristic
283C23—22no—
284C23—41no—
285C23—41no—
286C23#37911yescharacteristic
287C23#39111yescharacteristic
288C23#39111yescharacteristic
289Q8—88no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4#39111yescharacteristic
327C2×C4#39111yescharacteristic
328C2×C4#39111yescharacteristic
329C2×C4#39111yescharacteristic
330C2×C4#39311yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39311yescharacteristic
333C2×C4#39311yescharacteristic
334C2×C4#39511yescharacteristic
335C2×C4#39511yescharacteristic
336C2×C4#39511yescharacteristic
337C2×C4#39511yescharacteristic
338C8—82no—
339C8—82no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—82no—
351C22#144611yescharacteristic
352C22#179412yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—44no—
365C4—82no—
366C4—82no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4#203112yes—
371C2—22no—
372C2—41no—
373C2—41no—
374C2#1138711yescharacteristic
375C2#1139311yescharacteristic
376C2#1139511yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC22#1446
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ 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
42. #432Q8semidirect
43. #432Q8semidirect
51. #431Q8semidirect
55. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#346442
#81132
#80922
#80712
#80692
#79562
#79262