Finite Group Databasebeta

← #7572308

Subgroups — #7572308

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59924C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic
3#32695C214yesmaximal, complemented
4#29208C212yesmaximal, complemented
5#28816C214yesmaximal, complemented
6#26445C214yesmaximal, complemented
7#26040C214yesmaximal
8#25850C214yesmaximal
9#23250C211yesmaximal, characteristic, complemented
10#22969C214yesmaximal
11C4×#476C212yesmaximal
12C2×#554C2212yes—
13C2×#501C2214yes—
14C2×#476C2212yes—
15C2×#474C2214yes—
16#2868C2211yescharacteristic
17#2857—21no—
18#2811—22no—
19#2567—24no—
20#2567C2214yes—
21#2567C2212yes—
22#2552C2214yes—
23#2551—21no—
24#2551C2214yes—
25#2502—22no—
26#2440—24no—
27#2369—21no—
28#2369C2214yes—
29D4×C4⋊C4C2214yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2211yescharacteristic
32C2×#384—21no—
33C42×D4C2211yescharacteristic
34#2086C2218yes—
35#2059C2218yes—
36#2057C2218yes—
37#2057C2218yes—
38#1990C2214yes—
39#1989C2214yes—
40#1987C2212yes—
41#1985C2214yes—
42#1948C2212yes—
43#1947C2214yes—
44#1947C2214yes—
45#1944C2218yes—
46#1939C2214yes—
47#1920C2218yes—
48#1920C2214yes—
49#1864C2214yes—
50#1862C2218yes—
51#1845C2214yes—
52#1802C2212yes—
53C4×#106C2214yes—
54#1770C2214yes—
55C4×C4⋊C8C2212yes—
56#1762C2214yes—
57C4×#102C2214yes—
58#554—44no—
59#554—44no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2314yes—
66C2×#125C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—24no—
69C2×C4×Q8—22no—
70C2×C4×Q8C2312yes—
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2314yes—
73C2×C4×D4C2312yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—24no—
76C22×C42—21no—
77#501—48no—
78#476Q8116yescomplemented
79#474—48no—
80C2×#107C2318yes—
81C2×#107C2318yes—
82C2×#106C2314yes—
83C2×#106C2314yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2314yes—
86C2×C4⋊C8C2312yes—
87C2×#102C2314yes—
88C2×#102C2314yes—
89C2×#97C2314yes—
90C2×C4×C8C2314yes—
91#398—28no—
92#395—24no—
93#395—24no—
94#395—24no—
95#395—28no—
96#391—22no—
97#391—24no—
98#391—22no—
99#391—28no—
100#391D412yes—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—22no—
105#389D414yes—
106#389C2314yes—
107#389C2318yes—
108#389C2318yes—
109#385—28no—
110#384—24no—
111#384—22no—
112#384—24no—
113#384—21no—
114#384—21no—
115#384D411yescharacteristic
116#384C2312yes—
117#384C2314yes—
118#380—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2314yes—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2314yes—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2314yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C43C2311yescharacteristic
129#340C2314yes—
130#340C2314yes—
131#340C2318yes—
132#340C2318yes—
133#340C2314yes—
134#340C2314yes—
135#336C2314yes—
136#336C2318yes—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8C2×D412yes—
141C22×D4C2411yescharacteristic
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C22×C8C2414yes—
146C22×C8C2414yes—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—44no—
152C4⋊Q8—44no—
153C4⋊Q8—84no—
154C4⋊Q8—88no—
155C4⋊Q8C2×Q818yes—
156#125—28no—
157#125—44no—
158C22⋊Q8—416no—
159C22⋊Q8—48no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—84no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—28no—
166C4×D4—24no—
167C4×D4—44no—
168C4×D4C2×Q818yes—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—88no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42—44no—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204C2×C42C2414yes—
205C2×C42C2412yes—
206#107—48no—
207#107—48no—
208#106—28no—
209#106C2×Q8116yes—
210C4⋊C8—24no—
211C4⋊C8—44no—
212C4⋊C8C2×Q818yes—
213#102—28no—
214#102C2×Q8116yes—
215#97—44no—
216C4×C8—28no—
217#95—21no—
218#95—22no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—28no—
224#95C2×D418yes—
225C24—21no—
226C2×Q8—28no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—88no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4C22×Q814yes—
239C2×D4H(5,𝔽2)12yes—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—88no—
253C22×C4C22×D414yes—
254C22×C4C22×D414yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C2511yescharacteristic
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8C22×Q818yes—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—88no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—88no—
289C4⋊C4C22×Q818yes—
290C4⋊C4C22×Q814yes—
291C4⋊C4H(5,𝔽2)12yes—
292C22⋊C4—24no—
293C22⋊C4—28no—
294C22⋊C4—48no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—21no—
299C42—21no—
300C42—41no—
301C42—42no—
302C42—44no—
303C42—48no—
304C42—84no—
305C42#13714yes—
306C42C22×Q814yes—
307C422−1+414yes—
308C23—24no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23C23×D411yescharacteristic
314Q8—84no—
315Q8—88no—
316Q8—164no—
317Q8—168no—
318D4—24no—
319D4—22no—
320D4Q8214yes—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—28no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—416no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—168no—
349C2×C4D4×Q814yes—
350C2×C4D4×Q814yes—
351C2×C4C2×#13712yes—
352C2×C4C2×#13714yes—
353C2×C4#57614yes—
354C2×C4C23×Q812yes—
355C2×C4C2×H(5,𝔽2)11yescharacteristic
356C2×C4C2×H(5,𝔽2)11yescharacteristic
357C2×C4C2×2−1+412yes—
358C8—28no—
359C8—44no—
360C22—24no—
361C22—42no—
362C22—44no—
363C22—44no—
364C22—42no—
365C22C2×D4×Q812yes—
366C22#348411yescharacteristic
367C22C22×#13711yescharacteristic
368C22C2×#57612yes—
369C22#358611yescharacteristic
370C4—22no—
371C4—24no—
372C4—44no—
373C4—42no—
374C4—44no—
375C4—48no—
376C4—44no—
377C4—41no—
378C4—84no—
379C4—84no—
380C4#338518yes—
381C4C2×Q8211yescharacteristic
382C4#350312yes—
383C4#354411yescharacteristic
384C2—44no—
385C2#5915111yescharacteristic
386C2C2×#338512yes—
387C2#6138112yes—
388C2#6158611yescharacteristic
389C2#6166111yescharacteristic
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. #59924C2semidirect
3. #32695C2semidirect
4. #29208C2semidirect
5. #28816C2semidirect
6. #26445C2semidirect
9. #23250C2semidirect
78. #476Q8semidirect

Maximal subgroups

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

subgroupindex
#599242
#361802
#326952
#292082
#288162
#264452
#260402
#258502
#232502
#229692
C4×#4762