Finite Group Databasebeta

← #98796

Subgroups — #98796

Order 512.

774 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#391C211yesmaximal, characteristic
2#10795C212yesmaximal
3C2×#1272C212yesmaximal
4#7025C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#391C411yescharacteristic
7C2×#391C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C43C2211yescharacteristic
10C2×#336—22no—
11C2×#336—22no—
12C2×#336C2214yes—
13#1436—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16C4×#95—22no—
17#1272—24no—
18C23×Q8C2×C411yescharacteristic
19C23×C8—24no—
20C2×C4×Q8—22no—
21C2×C4×Q8—22no—
22C2×C4×Q8—42no—
23C2×C4×Q8—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33C22×C42D412yes—
34C22×C42C2311yescharacteristic
35#391—22no—
36#391—22no—
37#391C814yescomplemented
38#391D412yes—
39#391Q812yes—
40C4×C4⋊C4—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—21no—
45C2×#95—22no—
46C43—42no—
47#336—28no—
48#336—24no—
49#336—24no—
50#336—44no—
51C22×Q8—21no—
52C22×Q8—21no—
53C22×Q8—21no—
54C22×Q8—41no—
55C22×Q8—41no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8C4⋊C412yes—
59C22×Q8C2×C812yes—
60C22×Q8M1612yes—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D412yes—
70C22×C8—28no—
71C22×C8—44no—
72C22×C8—44no—
73C22×C8—44no—
74C22×C8—44no—
75C22×C8—44no—
76C22×C8—44no—
77C4×Q8—48no—
78C4×Q8—48no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4C4⋊C412yes—
92C2×C4⋊C4C2×C812yes—
93C2×C4⋊C4M1612yes—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42—24no—
103C2×C42—41no—
104C2×C42—42no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—44no—
113C2×C42C2×C812yes—
114C2×C42M1612yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42#4112yes—
118#95—28no—
119#95—24no—
120#95—24no—
121#95—44no—
122#95—42no—
123#95—44no—
124C24C2×C22⋊C411yescharacteristic
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8C4⋊C814yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C22⋊C814yes—
185C22×C4#9912yes—
186C22×C4#10012yes—
187C22×C4#10414yes—
188C22×C4C2×C4⋊C411yescharacteristic
189C22×C4#11711yescharacteristic
190C22×C4#12112yes—
191C22×C4C22⋊Q812yes—
192C22×C4C22×C811yescharacteristic
193C22×C4C2×M1611yescharacteristic
194C22×C4C2×M1611yescharacteristic
195C22×C4C2×M1611yescharacteristic
196C22×C4#13514yes—
197C22×C4#13714yes—
198C2×C8—48no—
199C2×C8—48no—
200C2×C8—48no—
201C2×C8—84no—
202C2×C8—84no—
203C2×C8—84no—
204C2×C8—84no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—48no—
212C42—24no—
213C42—24no—
214C42—42no—
215C42—42no—
216C42—41no—
217C42—41no—
218C42—42no—
219C42—48no—
220C42—48no—
221C42—48no—
222C42—84no—
223C23#38011yescharacteristic
224C23C2×C22⋊C811yescharacteristic
225C23#40711yescharacteristic
226C23C2×#9911yescharacteristic
227C23#41011yescharacteristic
228C23C2×#10011yescharacteristic
229C23#41311yescharacteristic
230C23#41612yes—
231C23#41912yes—
232C23C2×#10412yes—
233C23#42112yes—
234Q8—44no—
235Q8—44no—
236Q8—44no—
237Q8—44no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4#32914yes—
297C2×C4C2×C4⋊C811yescharacteristic
298C2×C4#42311yescharacteristic
299C2×C4#43111yescharacteristic
300C2×C4#43211yescharacteristic
301C2×C4#46414yes—
302C2×C4#47914yes—
303C8—88no—
304C22#145911yescharacteristic
305C22#146011yescharacteristic
306C22#146211yescharacteristic
307C22#146411yescharacteristic
308C22#147612yes—
309C22#147912yes—
310C22#148812yes—
311C22#149112yes—
312C22#150012yes—
313C22#150812yes—
314C22#151512yes—
315C22#151612yes—
316C22C2×#32911yescharacteristic
317C22#152912yes—
318C22#153611yescharacteristic
319C22#154311yescharacteristic
320C22#155412yes—
321C22#155811yescharacteristic
322C22#179711yescharacteristic
323C22#179911yescharacteristic
324C22#180011yescharacteristic
325C22#180512yes—
326C22#180612yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—24no—
332C4—24no—
333C4—24no—
334C4—44no—
335C4—44no—
336C4—42no—
337C4—82no—
338C4—82no—
339C4—82no—
340C2#756711yescharacteristic
341C2#758312yes—
342C2#760511yescharacteristic
343C2#761112yes—
344C2#762911yescharacteristic
345C2#764511yescharacteristic
346C2#855911yescharacteristic
347C2#856612yes—
348C2#857412yes—
349C2#857911yescharacteristic
350C2#859411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #391C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3912
#107952
C2×#12722
#70252