Finite Group Databasebeta

← #472164

Subgroups — #472164

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1942C211yesmaximal, characteristic
5#15153C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic
7#11379C212yesmaximal
8#11379C212yesmaximal
9#11373C212yesmaximal
10#11324C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#400—22no—
18C2×#395—22no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2212yes—
23#2099—24no—
24#2057—24no—
25#1990—24no—
26#1942—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1444—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—24no—
68C2×#107—22no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×#107Q814yescomplemented
73C2×C4×C8—42no—
74C2×C4×C8D414yes—
75#400—48no—
76#395—48no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#384—24no—
82#384—22no—
83#384—22no—
84#384—42no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#336—24no—
100#336—42no—
101#336D414yes—
102C22×Q8—44no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8—82no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4C2×D414yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8C2×D414yes—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C22×C8C2×Q812yes—
132C22×C8#4112yes—
133C4⋊Q8—28no—
134C4⋊Q8—48no—
135C4⋊Q8—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4#4114yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189#107—28no—
190#107—48no—
191#107—44no—
192C4×C8—44no—
193#95—24no—
194#95—42no—
195#95—42no—
196#95—48no—
197#95—48no—
198#95—48no—
199#95D818yes—
200C24C22×D411yescharacteristic
201C2×Q8—48no—
202C2×Q8—816no—
203C2×Q8—84no—
204C2×Q8—84no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4#12712yes—
243C22×C4#12712yes—
244C22×C4C2×D818yes—
245C22×C4C2×D814yes—
246C22×C4#13714yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×Q811yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4112yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+412yes—
254C22×C42−1+412yes—
255C22×C42−1+411yescharacteristic
256C22×C42−1+411yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8C22⋊Q814yes—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—48no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—816no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23C2×#12711yescharacteristic
309C23#53612yes—
310C23#54612yes—
311C23#54711yescharacteristic
312C23D4×Q811yescharacteristic
313C23C22×D812yes—
314C23C2×#13712yes—
315C23#57712yes—
316C23#57812yes—
317Q8—816no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—88no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4#49314yes—
359C2×C4#49714yes—
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4#52511yescharacteristic
362C2×C4#53411yescharacteristic
363C2×C4#53611yescharacteristic
364C2×C4#53611yescharacteristic
365C2×C4#53611yescharacteristic
366C2×C4#53612yes—
367C2×C4#53912yes—
368C2×C4#54112yes—
369C2×C4#54112yes—
370C2×C4#54412yes—
371C2×C4#54612yes—
372C2×C4#54711yescharacteristic
373C2×C4D4×Q811yescharacteristic
374C2×C4#55612yes—
375C2×C4#55712yes—
376C2×C4#56312yes—
377C2×C4#56312yes—
378C8—24no—
379C8—42no—
380C8—44no—
381C22#253911yescharacteristic
382C22#254211yescharacteristic
383C22#265811yescharacteristic
384C22#268912yes—
385C22#271312yes—
386C22C2×#49311yescharacteristic
387C22C2×#49711yescharacteristic
388C22#315411yescharacteristic
389C22#315611yescharacteristic
390C22#322614yes—
391C22#324214yes—
392C22#329414yes—
393C22#331214yes—
394C22#333412yes—
395C22#335612yes—
396C22Q8×D812yes—
397C22#338712yes—
398C4—24no—
399C4—24no—
400C4—48no—
401C4—82no—
402C4—82no—
403C4—84no—
404C4—162no—
405C4—161no—
406C4—161no—
407C4#258111yescharacteristic
408C4#262711yescharacteristic
409C4#263411yescharacteristic
410C4#268912yes—
411C4#271312yes—
412C4#281711yescharacteristic
413C2#1681311yescharacteristic
414C2#2832911yescharacteristic
415C2#2834511yescharacteristic
416C2#2852711yescharacteristic
417C2#2854211yescharacteristic
418C2#3027611yescharacteristic
419C2#3028811yescharacteristic
420C2#3142214yes—
421C2#3213014yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
72. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
C2×#20572
C2×#19902
C2×#19422
#151532
#151432
#113792
#113732
#113242
#107972