Finite Group Databasebeta

← #98252

Subgroups — #98252

Order 512.

937 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 937 conjugacy classes of subgroups — too many to draw. Only its 160 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
1D4×C4⋊Q8C211yesmaximal, characteristic
2#9896C212yesmaximal, complemented
3#8587C211yesmaximal, characteristic, complemented
4#8563C212yesmaximal
5#7571C211yesmaximal, characteristic, complemented
6C2×D4×Q8C412yes—
7C22×C4⋊Q8—21no—
8#2868C2211yescharacteristic
9#2857—21no—
10#2551—22no—
11#2546C412yes—
12#2440—24no—
13#2434—22no—
14D4×C4⋊C4—22no—
15C4×C4⋊Q8C2211yescharacteristic
16C42×D4C2211yescharacteristic
17#1842C2212yes—
18C4×C4⋊C8C2212yes—
19C2×#333—21no—
20C2×D4⋊C8—22no—
21#1296—21no—
22#1296—24no—
23#1279—22no—
24#1272—22no—
25C23×Q8—22no—
26D4×Q8—44no—
27D4×Q8—44no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—44no—
34C2×C4⋊Q8—42no—
35C2×C4⋊Q8—42no—
36C2×C4⋊Q8C2×C412yes—
37C2×C4⋊Q8D411yescharacteristic
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—24no—
40C2×C4×Q8C2×C412yes—
41C2×C4×D4—22no—
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4D412yes—
44C22×C4⋊C4—24no—
45C22×C42—21no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4×C8—22no—
51#398—24no—
52#395—28no—
53#391—22no—
54#391D412yes—
55#389—24no—
56#389—24no—
57#389—22no—
58#385—28no—
59#384—22no—
60#384D411yescharacteristic
61#384D414yes—
62#380—24no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4C2×C411yescharacteristic
65C4×C4⋊C4D412yes—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—22no—
68C43C2311yescharacteristic
69#336—22no—
70#333—42no—
71D4⋊C8—44no—
72C22×Q8—24no—
73C22×Q8—22no—
74C22×Q8—48no—
75C22×Q8—44no—
76C22×Q8—44no—
77C22×Q8—84no—
78C22×Q8C22⋊C412yes—
79C22×D4C22⋊C411yescharacteristic
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—24no—
83C22×C8—42no—
84C22×C8—42no—
85C4⋊Q8—22no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—48no—
89C4⋊Q8—48no—
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—42no—
97C4⋊Q8—82no—
98C4⋊Q8D812yes—
99C4⋊Q8SD1612yes—
100C22⋊Q8—416no—
101C22⋊Q8—48no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—44no—
109C4×D4—44no—
110C4×D4SD1614yes—
111C4×D4Q1614yes—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C22⋊C414yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—24no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C4⋊C8—24no—
141C4⋊C8—42no—
142C4⋊C8—44no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147C4×C8—42no—
148C4×C8—42no—
149#95—28no—
150#95—21no—
151#95—22no—
152#95—24no—
153C24—21no—
154C2×Q8—22no—
155C2×Q8—44no—
156C2×Q8—48no—
157C2×Q8—48no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—84no—
164C2×Q8—816no—
165C2×Q8—816no—
166C2×Q8#10214yes—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4#10012yes—
170C2×D4#10314yes—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—42no—
180C22×C4—48no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—48no—
188C22×C4—84no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4#12012yes—
193C22×C4#12012yes—
194C2×C8—44no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—416no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—84no—
216C4⋊C4#10012yes—
217C4⋊C4#10314yes—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—48no—
222C42—21no—
223C42—21no—
224C42—22no—
225C42—22no—
226C42—41no—
227C42—42no—
228C42—42no—
229C42—41no—
230C42—41no—
231C42—42no—
232C42—42no—
233C42C2×D811yescharacteristic
234C42C2×SD1611yescharacteristic
235C42C2×SD1612yes—
236C42C2×Q1612yes—
237C42AGL(1,ℤ/8)12yes—
238C42#13712yes—
239C42#13712yes—
240C23—22no—
241C23—24no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23#37911yescharacteristic
246Q8—44no—
247Q8—44no—
248Q8—84no—
249Q8—84no—
250Q8—816no—
251D4—24no—
252D4—22no—
253D4#33314yescomplemented
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—816no—
281C2×C4C2×#10011yescharacteristic
282C2×C4C2×#10011yescharacteristic
283C2×C4C2×#10211yescharacteristic
284C2×C4C2×#10312yes—
285C2×C4#41811yescharacteristic
286C2×C4#41912yes—
287C2×C4#44712yes—
288C2×C4#44812yes—
289C2×C4#44812yes—
290C2×C4#45012yes—
291C2×C4#45112yes—
292C2×C4#45112yes—
293C2×C4#45412yes—
294C2×C4#45612yes—
295C8—84no—
296C8—162no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—44no—
301C22—44no—
302C22#178311yescharacteristic
303C22#178611yescharacteristic
304C22#178712yes—
305C22#179312yes—
306C22#179511yescharacteristic
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—41no—
312C4—42no—
313C4—44no—
314C4—44no—
315C4—82no—
316C4—84no—
317C4C2×#33311yescharacteristic
318C4#154812yes—
319C4#155811yescharacteristic
320C4#162712yes—
321C4#163712yes—
322C4#164912yes—
323C4#165112yes—
324C2—44no—
325C2#834211yescharacteristic
326C2#838311yescharacteristic
327C2#842412yes—
328C2#844712yes—
329C2#847111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9896C2semidirect
3. #8587C2semidirect
5. #7571C2semidirect
253. D4#333semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#98962
#85872
#85632
#75712