Finite Group Databasebeta

← #7572306

Subgroups — #7572306

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59832C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic
3#32694C214yesmaximal, complemented
4#29206C212yesmaximal, complemented
5#28817C214yesmaximal, complemented
6#26447C214yesmaximal, complemented
7#26044C214yesmaximal
8#25851C214yesmaximal
9#23250C211yesmaximal, characteristic, complemented
10#22968C214yesmaximal
11C4×#474C212yesmaximal
12C2×Q82—21no—
13C2×#554C2212yes—
14C2×#501C2214yes—
15C2×#476C2214yes—
16C2×#474C2212yes—
17#2868C2211yescharacteristic
18#2857—21no—
19#2734—22no—
20#2582—21no—
21#2567C2212yes—
22#2552C2214yes—
23#2551C2214yes—
24#2546C2214yes—
25#2502—22no—
26#2451—24no—
27#2440—24no—
28#2365C2214yes—
29D4×C4⋊C4C2214yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2211yescharacteristic
32C2×#384—21no—
33C42×D4C2211yescharacteristic
34#2088C2218yes—
35#2059C2218yes—
36#2058C2218yes—
37#2058C2218yes—
38#1990C2214yes—
39#1989C2214yes—
40#1988C2212yes—
41#1985C2214yes—
42#1948C2214yes—
43#1948C2214yes—
44#1947C2212yes—
45#1943C2218yes—
46#1939C2214yes—
47#1921C2218yes—
48#1918C2214yes—
49#1864C2218yes—
50#1862C2214yes—
51#1844C2214yes—
52#1802C2212yes—
53C4×#107C2214yes—
54#1770C2214yes—
55C4×C4⋊C8C2212yes—
56#1762C2214yes—
57C4×#102C2214yes—
58Q82—48no—
59#554—44no—
60#554—44no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2314yes—
68C2×#125C2314yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—24no—
71C2×C4×Q8—24no—
72C2×C4×Q8—22no—
73C2×C4×Q8C2312yes—
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2314yes—
76C2×C4×D4C2312yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—24no—
79C22×C42—21no—
80#501—48no—
81#476—48no—
82#474Q8116yescomplemented
83C2×#107C2314yes—
84C2×#107C2314yes—
85C2×#106C2318yes—
86C2×#106C2318yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2314yes—
89C2×C4⋊C8C2312yes—
90C2×#102C2314yes—
91C2×#102C2314yes—
92C2×#97C2314yes—
93C2×C4×C8C2314yes—
94#398—24no—
95#398—24no—
96#398—28no—
97#395—24no—
98#395—28no—
99#391—22no—
100#391—24no—
101#391—24no—
102#391—22no—
103#391D412yes—
104#389—24no—
105#389—24no—
106#389—22no—
107#389D414yes—
108#389C2314yes—
109#389C2318yes—
110#389C2318yes—
111#387—28no—
112#385—28no—
113#384—22no—
114#384—24no—
115#384—21no—
116#384—21no—
117#384D411yescharacteristic
118#384C2312yes—
119#384C2314yes—
120#380—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2314yes—
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2314yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
130#340C2314yes—
131#340C2314yes—
132#340C2318yes—
133#340C2318yes—
134#340C2314yes—
135#340C2314yes—
136#336C2314yes—
137#336C2318yes—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—24no—
141C22×Q8C2×D412yes—
142C22×D4C2411yescharacteristic
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—24no—
146C22×C8C2414yes—
147C22×C8C2414yes—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C4⋊Q8—416no—
155C4⋊Q8—416no—
156C4⋊Q8—84no—
157C4⋊Q8C2×Q818yes—
158#125—28no—
159#125—44no—
160C22⋊Q8—416no—
161C22⋊Q8—48no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—416no—
165C4×Q8—48no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—28no—
169C4×D4—24no—
170C4×D4—44no—
171C4×D4C2×Q818yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—88no—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2412yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—44no—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2414yes—
208C2×C42C2412yes—
209#107—28no—
210#107C2×Q8116yes—
211#106—48no—
212#106—48no—
213C4⋊C8—24no—
214C4⋊C8—44no—
215C4⋊C8C2×Q818yes—
216#102—28no—
217#102C2×Q8116yes—
218#97—44no—
219C4×C8—28no—
220#95—21no—
221#95—22no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—28no—
226#95—28no—
227#95C2×D418yes—
228C24—21no—
229C2×Q8—28no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4C22×Q814yes—
243C2×D4H(5,𝔽2)12yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—88no—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D414yes—
259C22×C4C22×D414yes—
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C2511yescharacteristic
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8C22×Q818yes—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—416no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—82no—
290C4⋊C4—88no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4C22×Q818yes—
294C4⋊C4C22×Q814yes—
295C4⋊C4H(5,𝔽2)12yes—
296C22⋊C4—24no—
297C22⋊C4—28no—
298C22⋊C4—48no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—21no—
303C42—21no—
304C42—41no—
305C42—48no—
306C42—42no—
307C42—44no—
308C42—48no—
309C42AGL(1,ℤ/8)14yes—
310C42C22×Q814yes—
311C422−1+414yes—
312C23—24no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23C23×D411yescharacteristic
318Q8—416no—
319Q8—84no—
320Q8—168no—
321Q8—164no—
322D4—24no—
323D4—22no—
324D4Q8214yescomplemented
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—48no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—416no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—168no—
353C2×C4D4×Q814yes—
354C2×C4D4×Q814yes—
355C2×C4C2×AGL(1,ℤ/8)12yes—
356C2×C4C2×AGL(1,ℤ/8)14yes—
357C2×C4#57714yes—
358C2×C4C23×Q812yes—
359C2×C4C2×H(5,𝔽2)11yescharacteristic
360C2×C4C2×H(5,𝔽2)11yescharacteristic
361C2×C4C2×2−1+412yes—
362C8—28no—
363C8—44no—
364C22—24no—
365C22—42no—
366C22—44no—
367C22—44no—
368C22—42no—
369C22C2×D4×Q812yes—
370C22#348411yescharacteristic
371C22C22×AGL(1,ℤ/8)11yescharacteristic
372C22C2×#57712yes—
373C22#358611yescharacteristic
374C4—22no—
375C4—24no—
376C4—44no—
377C4—42no—
378C4—44no—
379C4—48no—
380C4—44no—
381C4—41no—
382C4—84no—
383C4—84no—
384C4#338418yes—
385C4C2×Q8211yescharacteristic
386C4#350312yes—
387C4#354411yescharacteristic
388C2—44no—
389C2#5915111yescharacteristic
390C2C2×#338412yes—
391C2#6138312yes—
392C2#6158511yescharacteristic
393C2#6166211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59832C2semidirect
3. #32694C2semidirect
4. #29206C2semidirect
5. #28817C2semidirect
6. #26447C2semidirect
9. #23250C2semidirect
82. #474Q8semidirect
324. D4Q82semidirect

Maximal subgroups

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

subgroupindex
#598322
#361802
#326942
#292062
#288172
#264472
#260442
#258512
#232502
#229682
C4×#4742