Finite Group Databasebeta

← #108672

Subgroups — #108672

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20524C211yesmaximal, characteristic
2#8114C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8009C211yesmaximal, characteristic
5#7943C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7C2×#1284C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1753—22no—
16#1753—22no—
17#1749C412yescomplemented
18C4×C22⋊C8C412yescomplemented
19#1306—22no—
20#1305—22no—
21#1293C414yescomplemented
22#1284—22no—
23#1284—22no—
24#1284C414yescomplemented
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C2×#117—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407C2×C412yes—
47#407C2×C412yes—
48#407C2×C412yes—
49#407D412yes—
50#407Q812yes—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8C2×C412yes—
59C2×#97—22no—
60C2×C4×C8—22no—
61#381—21no—
62#381—21no—
63#381—21no—
64#381—21no—
65#381C2×C412yes—
66#380—21no—
67#380—21no—
68#379—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4C2×C412yes—
73C4×C22⋊C4D412yes—
74C4×C22⋊C4D412yes—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78#336—22no—
79#336—22no—
80C25#4111yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—41no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×Q811yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—81no—
109C2×M16—81no—
110C2×M16—81no—
111C2×M16—81no—
112C2×M16—81no—
113C2×M16—81no—
114C2×M16—81no—
115C2×M16—81no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120#117—81no—
121#117—81no—
122C2×C4⋊C4—41no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—81no—
132C2×C22⋊C4—81no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—81no—
137C2×C42—81no—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C412yes—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8C4214yes—
155C22⋊C8C4⋊C414yes—
156#97—82no—
157C4×C8—82no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—41no—
164#95—41no—
165#95—41no—
166C24—42no—
167C24—41no—
168C24—41no—
169C24—81no—
170C24#11711yescharacteristic
171C24C4×D411yescharacteristic
172C24C4×Q811yescharacteristic
173C24#12311yescharacteristic
174C24#12311yescharacteristic
175C24#12611yescharacteristic
176C24#12611yescharacteristic
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4C2×C4211yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C4⋊C411yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4#11711yescharacteristic
215C22×C4#11711yescharacteristic
216C22×C4#11711yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×Q811yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12511yescharacteristic
235C22×C4#12511yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238M16—82no—
239M16—82no—
240M16—82no—
241M16—82no—
242M16—81no—
243M16—81no—
244M16—81no—
245M16—81no—
246M16—82no—
247M16—82no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—82no—
258C4⋊C4—82no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—82no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C42—42no—
269C42—42no—
270C42—81no—
271C42—81no—
272C42—82no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—84no—
298C23—84no—
299C23C4×C4⋊C411yescharacteristic
300C23#38111yescharacteristic
301C23#38311yescharacteristic
302C23#38711yescharacteristic
303C23#38711yescharacteristic
304C23#38811yescharacteristic
305C23#38811yescharacteristic
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—24no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
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—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—164no—
353C2×C4#37611yescharacteristic
354C2×C4C4×C22⋊C411yescharacteristic
355C2×C4#38011yescharacteristic
356C2×C4#38211yescharacteristic
357C2×C4#38711yescharacteristic
358C2×C4#38711yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C8—44no—
362C8—44no—
363C8—82no—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—44no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22#143411yescharacteristic
395C22#182811yescharacteristic
396C22#182811yescharacteristic
397C22#184112yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—82no—
408C4—162no—
409C4—162no—
410C2—21no—
411C2—21no—
412C2—22no—
413C2—41no—
414C2—41no—
415C2—41no—
416C2—41no—
417C2—41no—
418C2#1081011yescharacteristic
419C2#1081311yescharacteristic
420C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1749C4semidirect
18. C4×C22⋊C8C4semidirect
21. #1293C4semidirect
24. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#205242
#81142
#81082
#80092
#79432
#79422
C2×#12842