Finite Group Databasebeta

← #116771

Subgroups — #116771

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3#17748C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8121C211yesmaximal, characteristic
6#8098C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1989—22no—
16#1988—21no—
17#1987—21no—
18#1950—22no—
19#1855—22no—
20#1849—21no—
21#1838—21no—
22#1736—22no—
23#1558—21no—
24#1556—21no—
25#1554—22no—
26#1550—21no—
27#1550—21no—
28#1497C414yescomplemented
29#1491—22no—
30#1481—22no—
31#1307—22no—
32#1303—22no—
33#1285—22no—
34C22×M16D412yes—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432D412yes—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70C2×#108—24no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423D411yescharacteristic
77#423D411yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8D412yes—
82#407—21no—
83#407—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—21no—
87C4×M16—21no—
88C2×#97—21no—
89C2×#97—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C2×#95—21no—
95#341—24no—
96#340—24no—
97#340—24no—
98#333—22no—
99#333—22no—
100#332—22no—
101#332—22no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—22no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—44no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C4⋊C412yes—
125C4⋊Q8C4⋊C412yes—
126#125—22no—
127#125—22no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—82no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177#108—84no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8C22⋊C414yes—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—41no—
208#95—22no—
209#95—41no—
210C24#12411yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—81no—
216C2×Q8—81no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4#11711yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12611yescharacteristic
247M16—42no—
248M16—44no—
249M16—42no—
250M16—81no—
251M16—81no—
252M16—82no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—42no—
281C42—42no—
282C42—82no—
283C42—82no—
284C42C2×C22⋊C411yescharacteristic
285C42C2×C4⋊C411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42#12011yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C42#12611yescharacteristic
299C42#12711yescharacteristic
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23#38811yescharacteristic
305C23#39311yescharacteristic
306C23#39411yescharacteristic
307Q8—44no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—88no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#38011yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#39211yescharacteristic
348C2×C4#39411yescharacteristic
349C2×C4#39611yescharacteristic
350C2×C4#39611yescharacteristic
351C2×C4#39711yescharacteristic
352C2×C4#39911yescharacteristic
353C2×C4#45512yes—
354C2×C4#45612yes—
355C8—44no—
356C8—82no—
357C8—82no—
358C8—82no—
359C8—81no—
360C8—81no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—42no—
370C22#143911yescharacteristic
371C22#202812yes—
372C22#204811yescharacteristic
373C22#204911yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—84no—
385C4—82no—
386C4—82no—
387C4#188212yes—
388C4#200411yescharacteristic
389C4#200511yescharacteristic
390C2—21no—
391C2—21no—
392C2—22no—
393C2—41no—
394C2#1106411yescharacteristic
395C2#1111011yescharacteristic
396C2#1112011yescharacteristic
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
28. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#213122
#177482
#174962
#81212
#80982
C2×#12852