Finite Group Databasebeta

← #117252

Subgroups — #117252

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3#17648C211yesmaximal, characteristic
4#17584C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7938C211yesmaximal, characteristic
7#7895C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1950—22no—
16#1950—22no—
17#1810—22no—
18#1550—22no—
19#1550—22no—
20C2×#333C412yescomplemented
21C2×#332C412yescomplemented
22#1531—22no—
23#1528—22no—
24#1526—22no—
25#1518—22no—
26#1515—22no—
27#1508—22no—
28#1286—22no—
29#1283—22no—
30#1279—22no—
31C22×M16—21no—
32C22×M16—21no—
33C23×C8—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59C2×#108—24no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8D412yes—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407D412yes—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16D412yes—
81C4×M16D412yes—
82C2×#97—21no—
83C2×C4×C8—21no—
84#383—21no—
85#382D412yes—
86#381—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C22⋊C4—21no—
90#376—21no—
91#341—24no—
92#333—24no—
93#332—24no—
94#330—28no—
95#328—28no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—44no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—82no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—82no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4⋊C412yes—
122C4⋊Q8C4⋊C412yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170#108—84no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195#95—22no—
196#95—22no—
197#95—41no—
198#95—41no—
199C24#12411yescharacteristic
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×Q8—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4#11711yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233M16—24no—
234M16—42no—
235M16—44no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—82no—
241M16—82no—
242C2×C8—24no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—82no—
269C42—42no—
270C42—42no—
271C42—81no—
272C42—81no—
273C42C2×C22⋊C411yescharacteristic
274C42C2×C4⋊C411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42#12011yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42#12311yescharacteristic
286C42#12311yescharacteristic
287C42#12611yescharacteristic
288C42#12711yescharacteristic
289C23—22no—
290C23—22no—
291C23—41no—
292C23#38811yescharacteristic
293C23#39311yescharacteristic
294C23#39411yescharacteristic
295Q8—82no—
296Q8—82no—
297Q8—82no—
298Q8—81no—
299Q8—81no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#38011yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#39011yescharacteristic
335C2×C4#39211yescharacteristic
336C2×C4#39411yescharacteristic
337C2×C4#39611yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39711yescharacteristic
340C2×C4#39911yescharacteristic
341C2×C4#47112yes—
342C2×C4#47312yes—
343C8—44no—
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22#143911yescharacteristic
357C22#193212yes—
358C22#205212yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—162no—
371C4#190411yescharacteristic
372C4#190511yescharacteristic
373C4#197711yescharacteristic
374C4#197911yescharacteristic
375C4#206412yes—
376C2—22no—
377C2—22no—
378C2—41no—
379C2#1105711yescharacteristic
380C2#1112811yescharacteristic
381C2#1114611yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×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. C2×#333C4semidirect
21. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#219692
C2×#15502
#176482
#175842
#79602
#79382
#78952