Finite Group Databasebeta

← Q8×#557 #7700080

Subgroups — Q8×#557 #7700080

Order 512.

4995 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C219yesmaximal
2#59668C2118yesmaximal
3#36593C213yesmaximal
4#36563C219yesmaximal
5Q8×C4⋊Q8C213yesmaximal
6Q8×#125C216yesmaximal
7#36466C216yesmaximal
8C4×Q82C213yesmaximal
9C4×Q82C212yesmaximal
10C4×Q82C211yesmaximal, characteristic
11C4×#557C213yesmaximal
12C2×Q82C2213yes—
13C2×Q82C2211yescharacteristic
14C2×#557C2216yes—
15C2×#557C2211yescharacteristic
16#2868C2219yes—
17#2859C22118yes—
18#2806C22136yes—
19#2584C22136yes—
20#2582C22118yes—
21#2567C22118yes—
22#2552C22118yes—
23#2552C22118yes—
24#2552C22118yes—
25#2551C22118yes—
26#2551C22118yes—
27#2550C22136yes—
28#2546C22118yes—
29#2507C22136yes—
30#2507C22136yes—
31#2451C22136yes—
32#2370C22118yes—
33#2369C22118yes—
34#2369C2219yes—
35#2369C22118yes—
36#2369C2219yes—
37#2365C2219yes—
38#2355C2219yes—
39#2351C2216yes—
40Q8×C4⋊C4C2213yes—
41Q8×C4⋊C4C2216yes—
42Q8×C4⋊C4C2213yes—
43Q8×C4⋊C4C2211yescharacteristic
44Q8×C4⋊C4C2213yes—
45Q8×C4⋊C4C2216yes—
46Q8×C4⋊C4C2216yes—
47Q8×C4⋊C4C2219yes—
48Q8×C4⋊C4C2216yes—
49Q8×C4⋊C4C2216yes—
50C4×C4⋊Q8C22118yes—
51C4×C4⋊Q8C22118yes—
52C4×C4⋊Q8C2219yes—
53C4×C4⋊Q8C2219yes—
54C4×C4⋊Q8C2219yes—
55C4×#125C22118yes—
56C42×Q8C2216yes—
57C42×Q8C2219yes—
58C42×Q8C2213yes—
59C42×Q8C2216yes—
60C42×Q8C2213yes—
61Q82—224no—
62Q82Q8116yescomplemented
63#557—424no—
64#557Q8116yesdirect product factor
65C2×C4⋊Q8C2319yes—
66C2×C4⋊Q8C23118yes—
67C2×C4⋊Q8C23118yes—
68C2×C4⋊Q8C23118yes—
69C2×C4⋊Q8C2313yes—
70C2×C4⋊Q8C2319yes—
71C2×#125C23136yes—
72C2×#125C2316yes—
73C2×C4×Q8C2319yes—
74C2×C4×Q8C2313yes—
75C2×C4×Q8C2316yes—
76C2×C4×Q8C2312yes—
77C2×C4×Q8C2316yes—
78C2×C4×Q8C2313yes—
79C2×C4×Q8C2316yes—
80C2×C4×Q8C23118yes—
81C2×C4×Q8C2319yes—
82C2×C4×Q8C23112yes—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2313yes—
85C2×C4×Q8C2313yes—
86C2×C4×Q8C2312yes—
87C2×C4×Q8C2313yes—
88C2×C4×Q8C2311yescharacteristic
89#398C23172yes—
90#398C23172yes—
91#395C23172yes—
92#391C23118yes—
93#391C23136yes—
94#391C23136yes—
95#391C23118yes—
96#391C23118yes—
97#389C23118yes—
98#389C23118yes—
99#389C2316yes—
100#389C23136yes—
101#389C23118yes—
102#389C23118yes—
103#389C23136yes—
104#389C23118yes—
105#389C23136yes—
106#389C23118yes—
107#389C23118yes—
108#389C23118yes—
109#389C23118yes—
110#389C23136yes—
111#389C23136yes—
112#389C23136yes—
113#387C23136yes—
114#387C23136yes—
115#384C2313yes—
116#384C2319yes—
117#384C23118yes—
118#384C23118yes—
119#384C2319yes—
120#384C23118yes—
121#384C23118yes—
122#382C23118yes—
123C4×C4⋊C4C23118yes—
124C4×C4⋊C4C23118yes—
125C4×C4⋊C4C2319yes—
126C4×C4⋊C4C23118yes—
127C4×C4⋊C4C2319yes—
128C4×C4⋊C4C2319yes—
129C4×C4⋊C4C2319yes—
130C4×C4⋊C4C23118yes—
131C4×C4⋊C4C23118yes—
132C4×C4⋊C4C2319yes—
133C4×C4⋊C4C23118yes—
134C4×C4⋊C4C23118yes—
135C4×C4⋊C4C23118yes—
136C4×C4⋊C4C2319yes—
137C4×C4⋊C4C23118yes—
138C4×C4⋊C4C2313yes—
139C4×C4⋊C4C2319yes—
140C4×C4⋊C4C23118yes—
141C4×C4⋊C4C2319yes—
142C43C23118yes—
143C43C2319yes—
144C22×Q8C2413yes—
145C22×Q8C24118yes—
146C22×Q8C2416yes—
147C22×Q8C2411yescharacteristic
148C22×Q8C2411yescharacteristic
149C4⋊Q8—272no—
150C4⋊Q8—236no—
151C4⋊Q8—436no—
152C4⋊Q8—818no—
153C4⋊Q8C2×Q8124yes—
154C4⋊Q8C2×Q8172yes—
155#125—472no—
156#125C2×Q8148yes—
157C4×Q8—224no—
158C4×Q8—224no—
159C4×Q8—212no—
160C4×Q8—212no—
161C4×Q8—28no—
162C4×Q8—236no—
163C4×Q8—412no—
164C4×Q8—436no—
165C4×Q8—418no—
166C4×Q8—812no—
167C4×Q8C2×Q8116yes—
168C4×Q8C2×Q8124yes—
169C4×Q8C2×Q818yes—
170C4×Q8C2×Q8124yes—
171C4×Q8C2×Q8124yes—
172C4×Q8#4118yes—
173C2×C4⋊C4C2413yes—
174C2×C4⋊C4C2419yes—
175C2×C4⋊C4C24118yes—
176C2×C4⋊C4C2419yes—
177C2×C4⋊C4C24118yes—
178C2×C4⋊C4C24118yes—
179C2×C4⋊C4C2416yes—
180C2×C4⋊C4C2419yes—
181C2×C4⋊C4C2416yes—
182C2×C4⋊C4C24136yes—
183C2×C4⋊C4C24136yes—
184C2×C4⋊C4C24118yes—
185C2×C4⋊C4C24118yes—
186C2×C4⋊C4C24136yes—
187C2×C4⋊C4C24118yes—
188C2×C4⋊C4C2411yescharacteristic
189C2×C4⋊C4C2416yes—
190C2×C4⋊C4C2413yes—
191C2×C4⋊C4C2416yes—
192C2×C4⋊C4C2419yes—
193C2×C4⋊C4C2419yes—
194C2×C4⋊C4C2416yes—
195C2×C4⋊C4C2413yes—
196C2×C42C24118yes—
197C2×C42C2419yes—
198C2×C42C2419yes—
199C2×C42C24118yes—
200C2×C42C2416yes—
201C2×C42C2419yes—
202C2×C42C24118yes—
203C2×C42C24118yes—
204C2×C42C2419yes—
205C2×C42C2416yes—
206C2×C42C2413yes—
207C2×C42C2419yes—
208C2×C42C2413yes—
209#95C2416yes—
210#95C24118yes—
211#95C24118yes—
212#95C24118yes—
213#95C24136yes—
214#95C24118yes—
215#95C24136yes—
216#95C24136yes—
217C2×Q8—26no—
218C2×Q8—236no—
219C2×Q8—26no—
220C2×Q8—46no—
221C2×Q8—46no—
222C2×Q8—418no—
223C2×Q8—418no—
224C2×Q8C22×Q8112yes—
225C2×Q8C22×Q8124yes—
226C2×Q8C22×Q814yes—
227C2×Q8C22×Q814yes—
228C2×Q8C22×Q814yes—
229C2×Q8C2×#4114yes—
230C2×Q8H(5,𝔽2)14yes—
231C2×Q82−1+414yes—
232C22×C4C2516yes—
233C22×C4C2519yes—
234C22×C4C25118yes—
235C22×C4C2512yes—
236C22×C4C2513yes—
237C22×C4C2516yes—
238C22×C4C2519yes—
239C22×C4C2513yes—
240C22×C4C2513yes—
241C22×C4C2513yes—
242C22×C4C2511yescharacteristic
243C4⋊C4—236no—
244C4⋊C4—236no—
245C4⋊C4—218no—
246C4⋊C4—218no—
247C4⋊C4—236no—
248C4⋊C4—212no—
249C4⋊C4—218no—
250C4⋊C4—26no—
251C4⋊C4—418no—
252C4⋊C4—418no—
253C4⋊C4—46no—
254C4⋊C4—436no—
255C4⋊C4—436no—
256C4⋊C4—436no—
257C4⋊C4—418no—
258C4⋊C4C22×Q814yes—
259C4⋊C4C22×Q8124yes—
260C4⋊C4C22×Q8112yes—
261C4⋊C4C22×Q8124yes—
262C4⋊C4C22×Q8136yes—
263C4⋊C4C22×Q8136yes—
264C4⋊C4C22×Q8124yes—
265C4⋊C4C2×#41112yes—
266C42—236no—
267C42—236no—
268C42—218no—
269C42—212no—
270C42—218no—
271C42—218no—
272C42—418no—
273C42—418no—
274C42—49no—
275C42C22×Q8124yes—
276C42C22×Q8112yes—
277C42C22×Q8136yes—
278C42C2×#41112yes—
279C23C2611yescharacteristic
280Q8—224no—
281Q8—424no—
282Q8—836no—
283Q8#557116yesdirect product factor
284Q8Q82116yescomplemented
285C2×C4—26no—
286C2×C4—23no—
287C2×C4—22no—
288C2×C4—23no—
289C2×C4—29no—
290C2×C4—26no—
291C2×C4—22no—
292C2×C4—23no—
293C2×C4—29no—
294C2×C4—23no—
295C2×C4—218no—
296C2×C4—26no—
297C2×C4—29no—
298C2×C4—218no—
299C2×C4—26no—
300C2×C4—29no—
301C2×C4—29no—
302C2×C4—26no—
303C2×C4—218no—
304C2×C4C23×Q814yes—
305C2×C4C23×Q816yes—
306C2×C4C23×Q8112yes—
307C2×C4C23×Q8118yes—
308C2×C4C23×Q816yes—
309C2×C4C23×Q816yes—
310C2×C4C23×Q816yes—
311C2×C4C23×Q812yes—
312C2×C4C22×#4116yes—
313C2×C4C22×#4116yes—
314C2×C4C22×#4112yes—
315C2×C4C2×H(5,𝔽2)16yes—
316C2×C4C2×2−1+416yes—
317C2×C4#58512yes—
318C22C24×Q811yescharacteristic
319C22C24×Q811yescharacteristic
320C22C23×#4111yescharacteristic
321C22C22×H(5,𝔽2)11yescharacteristic
322C22C22×2−1+411yescharacteristic
323C22C2×#58511yescharacteristic
324C22H(7,𝔽2)11yescharacteristic
325C4—26no—
326C4—218no—
327C4—24no—
328C4—26no—
329C4—212no—
330C4—418no—
331C4—46no—
332C4—49no—
333C4C2×#557112yes—
334C4C2×Q82112yes—
335C4Q8×#4114yes—
336C2C22×#55711yescharacteristic
337C2C22×Q8211yescharacteristic
338C2C2×Q8×#4111yescharacteristic
339C2#6298611yescharacteristic
340C2#6298911yescharacteristic
341C2Q8×2−1+411yescharacteristic
342C2#6303911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
62. Q82Q8semidirect
64. #557Q8direct (N × H)
283. Q8#557direct (N × H)
284. Q8Q82semidirect

Maximal subgroups

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

subgroupindex
#598712
#596682
#365932
#365632
Q8×C4⋊Q82
Q8×#1252
#364662
C4×Q822
C4×#5572