Finite Group Databasebeta

← #363106

Subgroups — #363106

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×Q8C211yesmaximal, characteristic
2C2×#1985C212yesmaximal
3C2×#1940C212yesmaximal
4#11202C214yesmaximal
5#10832C211yesmaximal, characteristic
6#10832C212yesmaximal
7#10795C212yesmaximal
8#10693C211yesmaximal, characteristic
9C22×C4×Q8C412yes—
10C22×C4×Q8C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C4⋊C8C2214yes—
13C22×#97C2212yes—
14C42×Q8—28no—
15C2×C4×C4⋊C4C412yes—
16C2×C4×C4⋊C4C412yes—
17C2×C4×C4⋊C4C412yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C43C2212yes—
21C2×C43C2211yescharacteristic
22#1985—28no—
23#1940—28no—
24C2×#336C2214yes—
25C2×#336C2212yes—
26C2×#336C2214yes—
27C2×#336C2214yes—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C23×Q8C2×C411yescharacteristic
33C23×C8C2314yes—
34C23×C8C2312yes—
35C23×C8C2312yes—
36C2×C4×Q8—28no—
37C2×C4×Q8—28no—
38C2×C4×Q8—28no—
39C2×C4×Q8—24no—
40C2×C4×Q8—24no—
41C2×C4×Q8D414yes—
42C2×C4×Q8Q814yes—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C42C2×C412yes—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2×C412yes—
52C22×C42C2×C412yes—
53C22×C42C2×C412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2312yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—28no—
60C2×C4⋊C8—28no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—44no—
65C2×#97—24no—
66C2×#97—24no—
67C2×#97—24no—
68C4×C4⋊C4—28no—
69C4×C4⋊C4—28no—
70C4×C4⋊C4—28no—
71C4×C4⋊C4—28no—
72C4×C4⋊C4D418yes—
73C2×#95C2×C412yes—
74C2×#95C2×C412yes—
75C2×#95C2×C411yescharacteristic
76C2×#95C2×C411yescharacteristic
77C43—28no—
78C43—24no—
79#336—24no—
80#336—28no—
81#336—28no—
82#336—28no—
83#336—22no—
84#336—24no—
85#336—42no—
86#336Q814yescomplemented
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—24no—
90C22×Q8—24no—
91C22×Q8C4⋊C414yes—
92C23×C4C22×C412yes—
93C23×C4C22×C412yes—
94C23×C4C22×C412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C2411yescharacteristic
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8C2×Q814yes—
121C22×C8#4114yes—
122C4×Q8—264no—
123C4×Q8—216no—
124C4×Q8—216no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4C22⋊C418yes—
137C2×C4⋊C4C4⋊C414yes—
138C2×C4⋊C4C4⋊C418yes—
139C2×C4⋊C4M1618yes—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42M1618yes—
169C2×C42C2×D418yes—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2×D414yes—
173C2×C42C2×Q814yes—
174C2×C42C2×Q812yes—
175C2×C42C2×Q812yes—
176C2×C42#4112yes—
177C2×C42#4114yes—
178C4⋊C8—416no—
179C4⋊C8—48no—
180#97—48no—
181#95—24no—
182#95—24no—
183#95C22⋊C414yes—
184#95C22⋊C414yes—
185C24C23×C411yescharacteristic
186C2×Q8—216no—
187C2×Q8—28no—
188C2×Q8—28no—
189C2×Q8—28no—
190C2×Q8—216no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C4⋊C414yes—
230C22×C4C2×C4⋊C412yes—
231C22×C4C2×C4⋊C412yes—
232C22×C4C2×C4⋊C414yes—
233C22×C4#11712yes—
234C22×C4#11712yes—
235C22×C4C4×Q814yes—
236C22×C4C2×M1614yes—
237C22×C4C2×M1614yes—
238C22×C4C2×M1614yes—
239C22×C4C2×M1614yes—
240C22×C4C2×M1614yes—
241C22×C4C2×M1614yes—
242C22×C4#13114yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D412yes—
245C22×C4C22×Q811yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4112yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4112yes—
250C22×C4H(5,𝔽2)11yescharacteristic
251C22×C42−1+411yescharacteristic
252C22×C42−1+412yes—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C4⋊C4—216no—
275C4⋊C4—216no—
276C4⋊C4—28no—
277C4⋊C4—216no—
278C4⋊C4—216no—
279C42—216no—
280C42—216no—
281C42—28no—
282C42—28no—
283C42—28no—
284C42—28no—
285C42—24no—
286C42—24no—
287C42—216no—
288C42—48no—
289C42—48no—
290C42—48no—
291C42—42no—
292C42—42no—
293C42#12118yes—
294C42C22⋊Q818yes—
295C23C22×C22⋊C411yescharacteristic
296C23C22×C4⋊C411yescharacteristic
297C23C2×#11711yescharacteristic
298C23C2×C4×Q811yescharacteristic
299C23C4×#4111yescharacteristic
300C23#51911yescharacteristic
301C23#52011yescharacteristic
302C23C22×M1612yes—
303C23C22×M1612yes—
304C23C2×#13112yes—
305C23#56812yes—
306Q8—232no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4#38014yes—
355C2×C4#38014yes—
356C2×C4#40718yes—
357C2×C4#42318yes—
358C2×C4#44618yes—
359C2×C4C2×#12112yes—
360C2×C4C2×C22⋊Q812yes—
361C2×C4#52912yes—
362C2×C4#53312yes—
363C2×C4#54112yes—
364C2×C4#54212yes—
365C2×C4#54612yes—
366C2×C4#54712yes—
367C2×C4#54812yes—
368C2×C4#54812yes—
369C2×C4D4×Q811yescharacteristic
370C2×C4D4×Q812yes—
371C2×C4#55011yescharacteristic
372C2×C4#55712yes—
373C2×C4#55711yescharacteristic
374C2×C4Q8211yescharacteristic
375C8—44no—
376C8—44no—
377C8—84no—
378C22C2×#38011yescharacteristic
379C22#229511yescharacteristic
380C22#232011yescharacteristic
381C22#233911yescharacteristic
382C22Q8×C22⋊C411yescharacteristic
383C22Q8×C4⋊C411yescharacteristic
384C22#235111yescharacteristic
385C22C2×#40712yes—
386C22#288012yes—
387C22C2×#42312yes—
388C22#290612yes—
389C22#292011yescharacteristic
390C22#292011yescharacteristic
391C22#292211yescharacteristic
392C22#292311yescharacteristic
393C22C2×#44612yes—
394C22Q8×M1612yes—
395C22#296512yes—
396C22#297012yes—
397C22#298312yes—
398C22#298412yes—
399C22#298912yes—
400C22#299012yes—
401C4—24no—
402C4—28no—
403C4—24no—
404C4—28no—
405C4—28no—
406C4—44no—
407C4—44no—
408C4—44no—
409C4#242612yes—
410C4#242912yes—
411C4#245112yes—
412C4#245312yes—
413C2#1393511yescharacteristic
414C2#2062611yescharacteristic
415C2#2063011yescharacteristic
416C2#2134011yescharacteristic
417C2#2134611yescharacteristic
418C2#2202011yescharacteristic
419C2#2202411yescharacteristic
420C2#2247112yes—
421C2#2251012yes—
422C2#2272212yes—
423C2#2273612yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socle, prime radicalC24C23×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
86. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C42×Q82
C2×#19852
C2×#19402
#112022
#108322
#107952
#106932