Finite Group Databasebeta

← #134231

Subgroups — #134231

Order 512.

670 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#391C211yesmaximal, characteristic
2C2×#1319C212yesmaximal
3#8558C212yesmaximal
4#7189C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#391C411yescharacteristic
7C2×#391C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C43C411yescharacteristic
10#1797—22no—
11C2×#324C2214yes—
12#1436—22no—
13C4⋊#95C411yescharacteristic
14C4⋊#95C2211yescharacteristic
15C4×#95—22no—
16#1319—24no—
17#1280—22no—
18C23×Q8C2×C411yescharacteristic
19C2×C4×Q8—22no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8—42no—
23C2×C4×Q8C814yescomplemented
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2311yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D412yes—
35C2×C4⋊C8—42no—
36C2×C22⋊C8—24no—
37#391—22no—
38#391—22no—
39#391—22no—
40#391D412yes—
41#391Q812yes—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95D412yes—
49C43—42no—
50#336—42no—
51#324—24no—
52#324—24no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8C4⋊C412yes—
61C22×Q8C2×C812yes—
62C22×Q8M1612yes—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C22×C8—44no—
73C4×Q8—44no—
74C4×Q8—44no—
75C4×Q8—44no—
76C4×Q8—44no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C2×C812yes—
90C2×C4⋊C4C2×D411yescharacteristic
91C2×C4⋊C4C2×Q811yescharacteristic
92C2×C4⋊C4#4112yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—24no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—22no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—44no—
115C2×C42C4⋊C412yes—
116C2×C42C2×C812yes—
117C2×C42M1612yes—
118C2×C42M1612yes—
119C4⋊C8—84no—
120C22⋊C8—44no—
121C22⋊C8—44no—
122#95—28no—
123#95—24no—
124#95—24no—
125#95—42no—
126#95—44no—
127#95—44no—
128C24C2×C22⋊C411yescharacteristic
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
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×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C4⋊C814yescomplemented
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
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—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4C22⋊C812yes—
191C22×C4C22⋊C812yes—
192C22×C4#9911yescharacteristic
193C22×C4#9911yescharacteristic
194C22×C4#10012yes—
195C22×C4C2×C4⋊C411yescharacteristic
196C22×C4#11711yescharacteristic
197C22×C4#12112yes—
198C22×C4C22⋊Q812yes—
199C22×C4C22×C811yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4C2×M1611yescharacteristic
202C22×C4C2×M1611yescharacteristic
203C2×C8—84no—
204C2×C8—84no—
205C2×C8—84no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—48no—
214C42—22no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—48no—
223C42—48no—
224C42—44no—
225C42—44no—
226C42—44no—
227C42—82no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23#38011yescharacteristic
233C23C2×C22⋊C811yescharacteristic
234C23#40711yescharacteristic
235C23C2×#9911yescharacteristic
236C23#41011yescharacteristic
237C23C2×#10011yescharacteristic
238C23#41311yescharacteristic
239Q8—44no—
240Q8—44no—
241Q8—44no—
242Q8—44no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
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—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4#32312yes—
303C2×C4#32312yes—
304C2×C4#35412yes—
305C2×C4#35612yes—
306C2×C4C2×C4⋊C811yescharacteristic
307C2×C4#42311yescharacteristic
308C2×C4#43111yescharacteristic
309C2×C4#43211yescharacteristic
310C8—164no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22C2×#32311yescharacteristic
324C22#145811yescharacteristic
325C22#146511yescharacteristic
326C22#146611yescharacteristic
327C22#179711yescharacteristic
328C22#179911yescharacteristic
329C22#180011yescharacteristic
330C22C2×#35411yescharacteristic
331C22#212611yescharacteristic
332C22C2×#35611yescharacteristic
333C22#213211yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—44no—
340C4—44no—
341C4—44no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—82no—
346C4—82no—
347C4#131914yescomplemented
348C2—21no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2#855711yescharacteristic
353C2C2×#131911yescharacteristic
354C2#1143111yescharacteristic
355C2#1153111yescharacteristic
356C2#1153211yescharacteristic
357C2#1279811yescharacteristic
358C2#1280211yescharacteristic
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
23. C2×C4×Q8C8semidirect
144. C2×Q8C4⋊C8semidirect
347. C4#1319semidirect

Maximal subgroups

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

subgroupindex
C4×#3912
C2×#13192
#85582
#71892