Finite Group Databasebeta

← #495567

Subgroups — #495567

Order 512.

841 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×#3170C211yesmaximal, characteristic
2C2×C4×M32C211yesmaximal, characteristic
3#20826C211yesmaximal, characteristic
4#13290C212yesmaximal, complemented
5#13212C212yesmaximal, complemented
6#13066C214yesmaximal
7C4×#362C214yesmaximal, complemented
8C2×#533—21no—
9C22×M32C2212yes—
10#3170—21no—
11#3170—21no—
12#3170—21no—
13#3170—21no—
14#3170C412yes—
15#3170C412yes—
16#3170C412yes—
17#3170C412yes—
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#487—21no—
21C2×#467—21no—
22C2×#467C2212yes—
23C2×C4×Q16—21no—
24C2×C4×Q16C2212yes—
25#2920—21no—
26C22×C4×C8C2211yescharacteristic
27C2×#362C2218yes—
28C4×M32—21no—
29C4×M32—21no—
30C4×M32C2212yes—
31C4×M32C2212yes—
32C2×#346—21no—
33C2×C4×C16—21no—
34#1848C2212yes—
35#1837—21no—
36#1810—21no—
37#1810C2212yes—
38C4×#108—21no—
39C4×#108C2212yes—
40#1387C2218yes—
41C22×Q16—21no—
42C22×Q16C2312yes—
43C23×C8D411yescharacteristic
44C23×C8C2311yescharacteristic
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—41no—
50#533—41no—
51#533—41no—
52#533—41no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×#125—41no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—41no—
58C2×C4×Q8—22no—
59C2×C4×Q8—41no—
60C2×#117—41no—
61C22×C42D411yescharacteristic
62C2×M32—22no—
63C2×M32—28no—
64C2×M32C2314yes—
65C22×C16—22no—
66#500—21no—
67#500—21no—
68#500C2×C412yes—
69#500C2×C412yes—
70#494—21no—
71#494—21no—
72#494C2×C412yes—
73#494C2×C412yes—
74#487—22no—
75#487—22no—
76#487—22no—
77#487—22no—
78#467—22no—
79#467—22no—
80#467—22no—
81#467—22no—
82#467—22no—
83#467—22no—
84#467C2×C414yes—
85#467C2×C414yes—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16—22no—
90C4×Q16—22no—
91C4×Q16—22no—
92C4×Q16C2×C414yes—
93C4×Q16C2×C414yes—
94#432—41no—
95#432—41no—
96#431—41no—
97#431—41no—
98C2×#108—22no—
99C2×#108C2314yes—
100C2×#107—21no—
101C2×#107C2312yes—
102#423—22no—
103C2×#103—22no—
104C2×#103—22no—
105#407—22no—
106C4×M16—22no—
107C2×C4×C8C2×C411yescharacteristic
108C2×C4×C8C2×C411yescharacteristic
109C2×C4×C8C2×C411yescharacteristic
110C2×C4×C8C2×C411yescharacteristic
111C2×C4×C8D411yescharacteristic
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#362—216no—
120#346—22no—
121#346—22no—
122C4×C16—22no—
123C4×C16—22no—
124#341—22no—
125#341—22no—
126C22×Q8—22no—
127C22×Q8—41no—
128C23×C4SD1611yescharacteristic
129C23×C4SD1611yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—22no—
134C2×Q16—28no—
135C2×Q16—44no—
136C2×Q16C22×C414yes—
137C2×M16—24no—
138C22×C8—24no—
139C22×C8—24no—
140C22×C8C22⋊C411yescharacteristic
141C22×C8C22⋊C411yescharacteristic
142C22×C8C22⋊C411yescharacteristic
143C22×C8C22⋊C411yescharacteristic
144C22×C8C22×C411yescharacteristic
145C22×C8C22×C411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160#125—42no—
161#125—42no—
162#125—42no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C22⋊C4—42no—
191C2×C42—42no—
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42D811yescharacteristic
197C2×C42D811yescharacteristic
198C2×C42D811yescharacteristic
199C2×C42D811yescharacteristic
200C2×C42SD1611yescharacteristic
201C2×C42SD1611yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204M32—28no—
205M32—24no—
206M32#4118yes—
207C2×C16—28no—
208C2×C16—24no—
209#108—28no—
210#108—44no—
211#107—22no—
212#107—22no—
213#107—22no—
214#107C22×C414yes—
215C4⋊C8—42no—
216C4⋊C8—42no—
217#103—24no—
218#103—24no—
219#103—24no—
220#103—24no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223#97—42no—
224C4×C8—42no—
225C4×C8C22⋊C411yescharacteristic
226C4×C8C22⋊C411yescharacteristic
227C4×C8C22⋊C411yescharacteristic
228C4×C8C22⋊C411yescharacteristic
229C4×C8C22⋊C411yescharacteristic
230C4×C8C22⋊C411yescharacteristic
231C4×C8C22⋊C411yescharacteristic
232C4×C8C22⋊C411yescharacteristic
233C4×C8C22×C411yescharacteristic
234C4×C8C22×C411yescharacteristic
235C4×C8C22×C411yescharacteristic
236C4×C8C22×C411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C24C2×SD1611yescharacteristic
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—84no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#10211yescharacteristic
256C22×C4#10211yescharacteristic
257C22×C4#10211yescharacteristic
258C22×C4#10211yescharacteristic
259C22×C4#10211yescharacteristic
260C22×C4#10211yescharacteristic
261C22×C4#10211yescharacteristic
262C22×C4#10211yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×SD1611yescharacteristic
268C22×C4C2×SD1611yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×SD1611yescharacteristic
271C22×C4C2×SD1611yescharacteristic
272C22×C4C22×D411yescharacteristic
273Q16—28no—
274Q16—44no—
275Q16—44no—
276Q16—44no—
277M16—44no—
278M16—44no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—44no—
290C2×C8C2×C22⋊C411yescharacteristic
291C2×C8C2×C22⋊C411yescharacteristic
292C2×C8C2×C22⋊C411yescharacteristic
293C2×C8C2×C22⋊C411yescharacteristic
294C2×C8C2×C22⋊C411yescharacteristic
295C2×C8C2×C22⋊C411yescharacteristic
296C2×C8#13514yes—
297C2×C8C23×C411yescharacteristic
298C2×C8C22×D411yescharacteristic
299C2×C8C2×#4114yes—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C42—44no—
315C42—44no—
316C42#10211yescharacteristic
317C42#10211yescharacteristic
318C42#10211yescharacteristic
319C42#10211yescharacteristic
320C42#10211yescharacteristic
321C42#10211yescharacteristic
322C42#10211yescharacteristic
323C42#10211yescharacteristic
324C42C2×C22⋊C411yescharacteristic
325C42C2×C22⋊C411yescharacteristic
326C42C2×C22⋊C411yescharacteristic
327C42C2×C22⋊C411yescharacteristic
328C42C2×D811yescharacteristic
329C42C2×D811yescharacteristic
330C42C2×D811yescharacteristic
331C42C2×D811yescharacteristic
332C16—28no—
333C23—22no—
334C23—22no—
335C23—41no—
336C23C2×#10211yescharacteristic
337C23C2×#10211yescharacteristic
338C23C22×SD1611yescharacteristic
339Q8—44no—
340Q8—44no—
341Q8—84no—
342Q8—84no—
343Q8—84no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4#36214yescomplemented
367C2×C4C2×#10211yescharacteristic
368C2×C4C2×#10211yescharacteristic
369C2×C4C2×#10211yescharacteristic
370C2×C4C2×#10211yescharacteristic
371C2×C4C2×#10211yescharacteristic
372C2×C4C2×#10211yescharacteristic
373C2×C4C2×#10211yescharacteristic
374C2×C4C2×#10211yescharacteristic
375C2×C4C2×#10211yescharacteristic
376C2×C4C2×#10211yescharacteristic
377C2×C4C22×C22⋊C411yescharacteristic
378C2×C4#53212yes—
379C2×C4C22×D811yescharacteristic
380C2×C4C2×#13512yes—
381C8—21no—
382C8—21no—
383C8—21no—
384C8—21no—
385C8—84no—
386C8#41612yes—
387C8#41612yes—
388C8C4×#4112yes—
389C8#52512yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22C2×#36212yes—
399C22C22×#10211yescharacteristic
400C22#316712yes—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—84no—
409C4—84no—
410C4C2×#36212yes—
411C4C2×#36212yes—
412C4#214812yes—
413C4#229711yescharacteristic
414C4C2×#41611yescharacteristic
415C4#316912yes—
416C2—22no—
417C2—22no—
418C2—41no—
419C2#2140111yescharacteristic
420C2C22×#36211yescharacteristic
421C2C2×#214811yescharacteristic
subgroupG/H
centerC2×C4C2×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C22×#102
prime radicalC24C2×SD16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13290C2semidirect
5. #13212C2semidirect
7. C4×#362C2semidirect
366. C2×C4#362semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×C4×M322
#208262
#132902
#132122
#130662
C4×#3622