Finite Group Databasebeta

← #6651612

Subgroups — #6651612

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#33098C212yesmaximal
4#32695C214yesmaximal, complemented
5#32645C214yesmaximal, complemented
6#32595C218yesmaximal, complemented
7#29642C212yesmaximal
8#27611C212yesmaximal
9#24606C214yesmaximal
10#24064C212yesmaximal
11#21822C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#484C2214yes—
16C2×#432C2212yes—
17#2855C2212yes—
18#2855C2212yes—
19#2627—21no—
20#2627—21no—
21#2582C2214yes—
22#2581—22no—
23#2568C2212yes—
24#2567—22no—
25#2567—22no—
26#2567C2214yes—
27#2567C2212yes—
28#2567C2214yes—
29#2565—22no—
30C2×#395—21no—
31C2×#395—21no—
32#2317—21no—
33#2310—21no—
34C2×#391—21no—
35C2×#389C2212yes—
36C2×#389C2212yes—
37#2285C2211yescharacteristic
38C2×#384—21no—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2099C2214yes—
41#2086C2218yes—
42#2084C2214yes—
43#2066C2218yes—
44#2057C2214yes—
45#2057C2218yes—
46#2043C2214yes—
47#2024C2218yes—
48#1985C2214yes—
49#1985C2214yes—
50#1944C2218yes—
51#1935C2218yes—
52#1927C2218yes—
53#1920C2218yes—
54#1920C2218yes—
55#1872C2214yes—
56#1863C2218yes—
57#1853C2214yes—
58#1824C2214yes—
59#1817C2212yes—
60#1817C2212yes—
61C23×Q8—21no—
62C23×Q8D411yescharacteristic
63#533—44no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—44no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2314yes—
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—44no—
74C2×C4×Q8D412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8C2312yes—
77C2×C4×Q8C2312yes—
78C2×C4×Q8C2312yes—
79C2×#117—21no—
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#484—48no—
94#432—44no—
95C2×#107C2318yes—
96C2×#106C2318yes—
97C2×C4⋊C8C2314yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×C22⋊C8C2314yes—
101C2×#97C2314yes—
102#400—22no—
103#400—22no—
104#398—24no—
105#398—24no—
106#395—24no—
107#395—24no—
108#395—24no—
109#395—24no—
110#395—22no—
111#395—22no—
112#391—22no—
113#391—22no—
114#391—24no—
115#391—24no—
116#391—24no—
117#391C2314yes—
118#391C2314yes—
119#389—24no—
120#389—22no—
121#389—22no—
122#389—24no—
123#389C2314yes—
124#389C2318yes—
125#389C2314yes—
126#389C2314yes—
127#389C2314yes—
128#387—24no—
129#387—24no—
130#385—22no—
131#385—22no—
132#384—22no—
133#384—24no—
134#384—24no—
135#384C2312yes—
136#382—22no—
137#382—21no—
138#381—22no—
139#381—22no—
140#380—21no—
141#380C2312yes—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4C2314yes—
145C4×C4⋊C4C2312yes—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2318yes—
155#336C2314yes—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—21no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—84no—
163C22×Q8C2×D412yes—
164C22×Q8C2×D412yes—
165C22×Q8C2412yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8C2418yes—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—48no—
181#125—44no—
182C22⋊Q8—28no—
183C22⋊Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—44no—
191C4×Q8—44no—
192#117—44no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2412yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2414yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—44no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2412yes—
243C2×C42C2412yes—
244#107—48no—
245#106—216no—
246C4⋊C8—44no—
247#103—216no—
248#103—48no—
249C22⋊C8—28no—
250#97—44no—
251#95—24no—
252#95—28no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263C24C22×D411yescharacteristic
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—28no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—416no—
273C2×Q8—42no—
274C2×Q8—48no—
275C2×Q8—82no—
276C2×Q8—88no—
277C2×Q8—816no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4#13714yes—
304C22×C4#13714yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4H(5,𝔽2)14yes—
315C22×C4C2511yescharacteristic
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—28no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—88no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C42−1+418yes—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—44no—
358C42—48no—
359C42—44no—
360C23—22no—
361C23—22no—
362C23#53412yes—
363C23C2×#13712yes—
364C23C2×#13712yes—
365C23C23×D411yescharacteristic
366Q8—48no—
367Q8—48no—
368Q8—48no—
369Q8—84no—
370Q8—816no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—416no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—88no—
411C2×C4#53412yes—
412C2×C4C2×#13714yes—
413C2×C4C2×#13712yes—
414C2×C4C2×#13712yes—
415C2×C4C2×H(5,𝔽2)12yes—
416C2×C4C2×2−1+414yes—
417C8—48no—
418C22—22no—
419C22—24no—
420C22—24no—
421C22—22no—
422C22#319514yes—
423C22C2×#53411yescharacteristic
424C22#348612yes—
425C22C22×#13711yescharacteristic
426C22C22×#13711yescharacteristic
427C22#358512yes—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—42no—
436C4—41no—
437C4—48no—
438C4—84no—
439C4—82no—
440C4—84no—
441C4—82no—
442C4#354414yes—
443C2—24no—
444C2#3725811yescharacteristic
445C2C2×#319511yescharacteristic
446C2#6075311yescharacteristic
447C2#6182612yes—
448C2#6183012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32695C2semidirect
5. #32645C2semidirect
6. #32595C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#25672
#330982
#326952
#326452
#325952
#296422
#276112
#246062
#240642
#218222