Finite Group Databasebeta

← #117395

Subgroups — #117395

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic, complemented
2#17714C211yesmaximal, characteristic
3#17544C211yesmaximal, characteristic
4C2×#1484C211yesmaximal, characteristic
5#8068C211yesmaximal, characteristic, complemented
6#7936C211yesmaximal, characteristic, complemented
7#7866C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17C4×#108—22no—
18#1551—22no—
19#1548C411yescharacteristic, complemented
20#1548C411yescharacteristic, complemented
21#1545C411yescharacteristic, complemented
22#1545C411yescharacteristic, complemented
23#1543—22no—
24#1507—22no—
25#1505—22no—
26#1502—22no—
27#1484—22no—
28#1484—22no—
29C2×D4⋊C8—22no—
30#1300—22no—
31#1283—22no—
32#1276—22no—
33C22×#41—21no—
34C22×M16—21no—
35#532—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532D412yes—
42#532Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C411yescharacteristic
67#431C2×C411yescharacteristic
68#431D412yes—
69C2×#108—24no—
70#423—21no—
71#423C2×C411yescharacteristic
72#423C2×C411yescharacteristic
73C2×C4⋊C8D412yes—
74C2×C4⋊C8D412yes—
75#407—21no—
76#407D412yes—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16D412yes—
85C2×#97—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93#376D412yes—
94C43—21no—
95#341—24no—
96#333—22no—
97#333—22no—
98#331—22no—
99#331—22no—
100#327—28no—
101D4⋊C8—28no—
102C2×#41—22no—
103C2×#41—22no—
104C2×#41—41no—
105C2×#41—41no—
106C2×#41—84no—
107C22×Q8—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—41no—
112C23×C4—41no—
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—22no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—22no—
120C2×M16—44no—
121C2×M16—41no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8C4⋊C412yes—
129#127—21no—
130#127—21no—
131#127—21no—
132#127C4⋊C412yes—
133#124—22no—
134#124—22no—
135#124—41no—
136#124—41no—
137#121—22no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—41no—
142#121—41no—
143#121—41no—
144#121—41no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153#117—24no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176#108—48no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8C22⋊C412yes—
187C4⋊C8C22⋊C412yes—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—41no—
199#97—41no—
200#97—41no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—41no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—22no—
212C24—41no—
213C24#12411yescharacteristic
214#41—84no—
215#41—84no—
216#41—84no—
217#41—84no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4—84no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—84no—
243C22×C4#11711yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261M16—24no—
262M16—42no—
263M16—44no—
264M16—44no—
265M16—42no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—82no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—24no—
288C42—42no—
289C42—82no—
290C42—82no—
291C42C2×C22⋊C411yescharacteristic
292C42C2×C4⋊C411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42#12011yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42#12311yescharacteristic
304C42#12311yescharacteristic
305C42#12611yescharacteristic
306C42#12711yescharacteristic
307C23—22no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23#38811yescharacteristic
315C23#39311yescharacteristic
316C23#39411yescharacteristic
317Q8—82no—
318Q8—82no—
319D4—44no—
320D4—82no—
321D4—82no—
322D4—82no—
323D4—84no—
324D4—84no—
325D4—82no—
326D4—81no—
327D4—81no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4#38011yescharacteristic
355C2×C4#38511yescharacteristic
356C2×C4#38511yescharacteristic
357C2×C4#38611yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#39011yescharacteristic
360C2×C4#39211yescharacteristic
361C2×C4#39411yescharacteristic
362C2×C4#39611yescharacteristic
363C2×C4#39611yescharacteristic
364C2×C4#39711yescharacteristic
365C2×C4#39911yescharacteristic
366C2×C4#44312yes—
367C2×C4#44412yes—
368C8—44no—
369C8—42no—
370C8—42no—
371C8—82no—
372C8—82no—
373C8—81no—
374C8—81no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—164no—
388C22#143911yescharacteristic
389C22#192912yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—84no—
399C4—82no—
400C4#186612yes—
401C4#190612yes—
402C4#197411yescharacteristic
403C4#197511yescharacteristic
404C4#204812yes—
405C2—22no—
406C2—41no—
407C2—41no—
408C2—162no—
409C2#1106011yescharacteristic
410C2#1112511yescharacteristic
411C2#1114211yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#532C22
LCSG ▷ C2×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
1. #21370C2semidirect
5. #8068C2semidirect
6. #7936C2semidirect
7. #7866C2semidirect
19. #1548C4semidirect
20. #1548C4semidirect
21. #1545C4semidirect
22. #1545C4semidirect

Maximal subgroups

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

subgroupindex
#213702
#177142
#175442
C2×#14842
#80682
#79362
#78662