Finite Group Databasebeta

← #6355377

Subgroups — #6355377

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2C2×#3113C212yesmaximal
3C2×#2984C211yesmaximal, characteristic
4#19476C218yesmaximal, complemented
5#19471C218yesmaximal, complemented
6#19290C214yesmaximal, complemented
7#19265C214yesmaximal, complemented
8#17818C211yesmaximal, characteristic
9#17755C212yesmaximal
10C2×#533C2212yes—
11C22×C4×Q8C2211yescharacteristic
12#3249—22no—
13#3230C2214yes—
14#3209C2214yes—
15#3189—21no—
16#3189—21no—
17#3170C2214yes—
18#3113—22no—
19#3113—22no—
20#3113—24no—
21#3113C2218yes—
22#3113C2214yes—
23#3113C2214yes—
24#3104—24no—
25C2×#483—22no—
26C2×#475C2214yes—
27C2×#462C2214yes—
28C2×#451—22no—
29#2984—24no—
30#2984C2218yes—
31C2×#446—22no—
32C2×#446C2212yes—
33C2×C8×Q8C2212yes—
34C2×#441—24no—
35C2×#432—21no—
36C2×#432C2211yescharacteristic
37C2×#432C2212yes—
38C2×#431—21no—
39C2×#431C2211yescharacteristic
40#1700C22116yescomplemented
41#1694C2218yescomplemented
42#1690C2218yescomplemented
43#1680C22116yescomplemented
44#1674C2218yescomplemented
45#1670C2218yescomplemented
46#1573C2218yes—
47#1568C2214yes—
48#1556C2214yes—
49#1556C2214yes—
50#1552C2218yes—
51#1552C2218yes—
52C23×Q8D411yescharacteristic
53C22×Q16—24no—
54#533—22no—
55#533—22no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533C2314yes—
61#533C2314yes—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2312yes—
64C2×#125—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D414yes—
76C2×C4×Q8C2314yes—
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×#117—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#500C2314yes—
85#494C2314yes—
86#487C2318yes—
87#484—24no—
88#484—24no—
89#483—28no—
90#483—24no—
91#483—24no—
92#477—28no—
93#475—24no—
94#475—24no—
95#475C2318yes—
96#475C2318yes—
97#467—24no—
98#467—24no—
99#462—24no—
100#462—24no—
101#462—28no—
102#462C2318yes—
103#462C2318yes—
104#451—24no—
105#451—28no—
106#451—24no—
107#451—24no—
108#451—24no—
109#446—216no—
110#446—24no—
111#446C2318yes—
112C8×Q8—24no—
113C8×Q8C2318yes—
114#441—28no—
115#441—28no—
116#441—28no—
117#441—28no—
118C4×Q16—24no—
119C4×Q16—24no—
120#432—22no—
121#432—22no—
122#432—22no—
123#432—28no—
124#432C2314yes—
125#432C2314yes—
126#432C2314yes—
127#431—28no—
128#431D414yes—
129#431C2314yes—
130C2×#106—24no—
131C2×C4⋊C8—22no—
132C2×C4⋊C8—22no—
133C2×C4⋊C8D412yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136C2×#103—24no—
137C2×#103—24no—
138C2×C22⋊C8—22no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—22no—
143C2×#97—22no—
144C2×#97—21no—
145C2×C4×C8—22no—
146C2×C4×C8C2312yes—
147#335C2318yes—
148#334C2318yes—
149#332C23116yes—
150#332C23116yes—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8C2×D411yescharacteristic
157C22×Q8C2×D414yes—
158C22×Q8C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×Q16—28no—
165C2×Q16—28no—
166C2×Q16—24no—
167C2×Q16—24no—
168C2×Q16—416no—
169C22×C8—24no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8C2414yes—
180C4⋊Q8C2414yes—
181#125—24no—
182#125—24no—
183#125—42no—
184#125—42no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C4×Q8—24no—
194C4×Q8—22no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—24no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8C2×D412yes—
212C4×Q8C2×D412yes—
213C4×Q8C2×D414yes—
214C4×Q8C2×D414yes—
215C4×Q8C2412yes—
216C4×Q8C2412yes—
217#117—24no—
218#117—24no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—42no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—24no—
245C2×C42—42no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4112yes—
251C2×C42#4111yescharacteristic
252C2×C42C2412yes—
253C2×C42C2411yescharacteristic
254#107—24no—
255#107—24no—
256#106—28no—
257#106—28no—
258C4⋊C8—28no—
259C4⋊C8—28no—
260C4⋊C8—22no—
261C4⋊C8—24no—
262C4⋊C8C2×D418yes—
263C4⋊C8C2414yes—
264C4⋊C8C2418yes—
265#103—28no—
266#103—28no—
267#103—28no—
268#103—28no—
269#103—28no—
270#103—28no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—24no—
274C22⋊C8C2×D414yes—
275C22⋊C8C2×D414yes—
276#97—28no—
277#97—24no—
278#97—24no—
279#97—24no—
280C4×C8—28no—
281C4×C8C2×D414yes—
282C4×C8C2414yes—
283C24#12111yescharacteristic
284C2×Q8—24no—
285C2×Q8—21no—
286C2×Q8—24no—
287C2×Q8—21no—
288C2×Q8—28no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—44no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—84no—
299C2×Q8C22×D412yes—
300C2×Q8C22×D412yes—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4#12112yes—
315C22×C4#12111yescharacteristic
316C22×C4#12112yes—
317C22×C4#12111yescharacteristic
318C22×C4#12112yes—
319C22×C4#12111yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4112yes—
325C22×C4C2×#4111yescharacteristic
326Q16—416no—
327Q16—416no—
328Q16—84no—
329Q16—84no—
330C2×C8—24no—
331C2×C8—28no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—48no—
336C2×C8C22×D414yes—
337C2×C8C22×D414yes—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4C22×D412yes—
360C4⋊C4C22×D412yes—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C42—22no—
364C42—24no—
365C42—22no—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—44no—
370C42—44no—
371C42#12112yes—
372C42#12112yes—
373C42#12112yes—
374C42C22×D411yescharacteristic
375C42C22×D411yescharacteristic
376C42C22×D412yes—
377C42C22×D412yes—
378C42C2×#4111yescharacteristic
379C42C2×#4112yes—
380C42C2511yescharacteristic
381C23—41no—
382C23—41no—
383C23#47314yes—
384C23C2×#12111yescharacteristic
385C23C2×#12112yes—
386Q8—24no—
387Q8—22no—
388Q8—24no—
389Q8—48no—
390Q8—42no—
391Q8—48no—
392Q8—48no—
393Q8—84no—
394Q8—84no—
395Q8#57514yes—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12112yes—
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12112yes—
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12112yes—
428C2×C4C23×D411yescharacteristic
429C2×C4C23×D411yescharacteristic
430C2×C4C22×#4111yescharacteristic
431C8—24no—
432C8—24no—
433C8—44no—
434C8—48no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—44no—
440C22C2×#47312yes—
441C22C2×#47314yes—
442C22C22×#12111yescharacteristic
443C4—21no—
444C4—22no—
445C4—24no—
446C4—21no—
447C4—22no—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—84no—
455C4—84no—
456C4C2×#57512yes—
457C4#358612yes—
458C2—24no—
459C2—22no—
460C2#6042411yescharacteristic
461C2#6044411yescharacteristic
462C2C22×#47311yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #19476C2semidirect
5. #19471C2semidirect
6. #19290C2semidirect
7. #19265C2semidirect
40. #1700C22semidirect
41. #1694C22semidirect
42. #1690C22semidirect
43. #1680C22semidirect
44. #1674C22semidirect
45. #1670C22semidirect

Maximal subgroups

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

subgroupindex
#609432
C2×#31132
C2×#29842
#194762
#194712
#192902
#192652
#178182
#177552