Finite Group Databasebeta

← #134233

Subgroups — #134233

Order 512.

608 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13786C211yesmaximal, characteristic
2C2×#1319C212yesmaximal
3#8558C212yesmaximal
4#7189C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#391C411yescharacteristic
7C2×#391C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×#376C411yescharacteristic
10#1797—22no—
11C2×#324C2214yes—
12#1436—22no—
13C4⋊#95C411yescharacteristic
14C4⋊#95C2211yescharacteristic
15#1434—22no—
16#1319—24no—
17#1280—22no—
18C23×Q8C2×C411yescharacteristic
19C2×C4×Q8—22no—
20C2×C4×Q8—42no—
21C2×C4×Q8—42no—
22C2×C4×Q8C814yescomplemented
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2311yescharacteristic
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32C2×C4⋊C8—42no—
33C2×C22⋊C8—24no—
34#391—22no—
35#391—22no—
36#391—22no—
37#391D412yes—
38#391Q812yes—
39C4×C4⋊C4—42no—
40#376—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95D412yes—
48#336—42no—
49#324—24no—
50#324—24no—
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—44no—
71C4×Q8—48no—
72C4×Q8—48no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4C2×C812yes—
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4C2×Q811yescharacteristic
88C2×C4⋊C4#4112yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42—42no—
101C2×C42—42no—
102C2×C42—82no—
103C2×C42C4⋊C412yes—
104C2×C42C2×C812yes—
105C2×C42M1612yes—
106C2×C42M1612yes—
107C4⋊C8—84no—
108C22⋊C8—44no—
109C22⋊C8—44no—
110#95—28no—
111#95—24no—
112#95—24no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—82no—
117#95—82no—
118C24C2×C22⋊C411yescharacteristic
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8C4⋊C814yescomplemented
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4C22⋊C812yes—
181C22×C4C22⋊C812yes—
182C22×C4#9912yes—
183C22×C4#10012yes—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4#11711yescharacteristic
186C22×C4#12112yes—
187C22×C4C22⋊Q812yes—
188C22×C4C22×C811yescharacteristic
189C22×C4C2×M1611yescharacteristic
190C22×C4C2×M1611yescharacteristic
191C22×C4C2×M1611yescharacteristic
192C2×C8—84no—
193C2×C8—84no—
194C2×C8—84no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—48no—
203C42—24no—
204C42—24no—
205C42—42no—
206C42—42no—
207C42—44no—
208C42—42no—
209C42—42no—
210C42—48no—
211C42—82no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23#38011yescharacteristic
217C23C2×C22⋊C811yescharacteristic
218C23#40711yescharacteristic
219C23C2×#9911yescharacteristic
220C23#41011yescharacteristic
221C23C2×#10011yescharacteristic
222C23#41311yescharacteristic
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—44no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4#32314yes—
287C2×C4C2×C4⋊C811yescharacteristic
288C2×C4#42311yescharacteristic
289C2×C4#43111yescharacteristic
290C2×C4#43211yescharacteristic
291C8—164no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22C2×#32311yescharacteristic
305C22#145811yescharacteristic
306C22#146511yescharacteristic
307C22#146611yescharacteristic
308C22#179711yescharacteristic
309C22#179911yescharacteristic
310C22#180011yescharacteristic
311C22#212711yescharacteristic
312C22#212911yescharacteristic
313C22#213311yescharacteristic
314C22#213511yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—24no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—42no—
324C4—82no—
325C4—82no—
326C4—82no—
327C4—82no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#855711yescharacteristic
333C2#1143211yescharacteristic
334C2#1143411yescharacteristic
335C2#1153511yescharacteristic
336C2#1153611yescharacteristic
337C2#1280011yescharacteristic
338C2#1280411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. C2×C4×Q8C8semidirect
134. C2×Q8C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#137862
C2×#13192
#85582
#71892