Finite Group Databasebeta

← #7617320

Subgroups — #7617320

Order 512.

1297 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45364C212yesmaximal
2#32694C212yesmaximal
3#32691C212yesmaximal
4#29293C212yesmaximal
5#28812C212yesmaximal
6#28812C212yesmaximal
7#28679C212yesmaximal
8#28676C211yesmaximal, characteristic
9#28675C211yesmaximal, characteristic
10#26455C214yesmaximal
11#26454C214yesmaximal
12#23158C212yesmaximal
13#23157C212yesmaximal
14#21725C212yesmaximal
15#21715C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2212yes—
19C2×#499C2212yes—
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22#2811—22no—
23#2806—24no—
24#2792—22no—
25#2586—22no—
26#2584C2214yes—
27#2582—22no—
28#2567C2212yes—
29#2550C2214yes—
30#2550C2214yes—
31#2546C2212yes—
32#2451—24no—
33#2399C2211yescharacteristic
34#2369C2214yes—
35#2365C2214yes—
36#2331C2214yes—
37#2088C2214yes—
38#2088C2214yes—
39#2087C2212yes—
40#2087C2212yes—
41#2086C2212yes—
42#2086C2212yes—
43#2059C2214yes—
44#2059C2214yes—
45#2058C2214yes—
46#2058C2214yes—
47#1993C2214yes—
48#1989C2214yes—
49#1988C2212yes—
50#1988C2214yes—
51#1987C2212yes—
52#1987C2214yes—
53#1985C2211yescharacteristic
54#1985C2211yescharacteristic
55#1949C2214yes—
56#1949C2214yes—
57#1943C2214yes—
58#1942C2212yes—
59#1942C2212yes—
60#1942C2212yes—
61#1942C2212yes—
62#1942C2212yes—
63#1942C2212yes—
64#1940C2212yes—
65#1940C2212yes—
66#1921C2214yes—
67#1921C2214yes—
68#1918C2214yes—
69#1918C2214yes—
70#1846C2212yes—
71#1843C2211yescharacteristic
72#1776C2214yes—
73C4×#107C2211yescharacteristic
74C4×#107C2211yescharacteristic
75C4×#107C2212yes—
76C4×#107C2212yes—
77C4×#107C2212yes—
78C8×C4⋊C4C2212yes—
79C8×C4⋊C4C2212yes—
80#1726C2211yescharacteristic
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8C2314yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125C2314yes—
86C2×C4×Q8—24no—
87C2×C4×Q8C2311yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89#500—42no—
90#500—42no—
91#500Q8116yescomplemented
92#499—44no—
93C8×Q8—42no—
94C8×Q8—42no—
95C2×#107C2312yes—
96C2×#107C2312yes—
97C2×#107C2312yes—
98C2×#107C2312yes—
99C2×#107C2314yes—
100C2×#107C2312yes—
101C2×#107C2312yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111C2×#97C2312yes—
112C2×#97C2311yescharacteristic
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2312yes—
118#398—24no—
119#398—22no—
120#398—24no—
121#398—24no—
122#398—22no—
123#398D411yescharacteristic
124#398D411yescharacteristic
125#398D412yes—
126#395—24no—
127#395—22no—
128#395—22no—
129#395—24no—
130#395—24no—
131#395D412yes—
132#395D411yescharacteristic
133#395D411yescharacteristic
134#391—22no—
135#391—22no—
136#391—24no—
137#391—24no—
138#391—24no—
139#391—24no—
140#391—24no—
141#391—24no—
142#391—24no—
143#389—24no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389—24no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#387—24no—
155#387—24no—
156#387—24no—
157#387—24no—
158#384—22no—
159#382C2314yes—
160C4×C4⋊C4—24no—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2312yes—
166C4×C4⋊C4C2312yes—
167C4×C4⋊C4C2312yes—
168#376C2311yescharacteristic
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2312yes—
176#340C2314yes—
177#340C2312yes—
178#340C2314yes—
179#340C2314yes—
180#336C2312yes—
181#336C2312yes—
182#336C2312yes—
183#336C2312yes—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8C2×D411yescharacteristic
188C22×Q8C2×D411yescharacteristic
189C22×C8C2412yes—
190C22×C8C2411yescharacteristic
191C22×C8C2411yescharacteristic
192C22×C8C2411yescharacteristic
193C22×C8C2412yes—
194C22×C8C2411yescharacteristic
195C4⋊Q8—28no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8—84no—
201C4⋊Q8C2×Q8116yes—
202#125—44no—
203#125—44no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—88no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42C2411yescharacteristic
239C2×C42C2412yes—
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2412yes—
243#107—24no—
244#107—24no—
245#107—42no—
246#107—42no—
247#107C2×Q8116yes—
248#107C2×Q818yes—
249#107C2×Q818yes—
250#106—44no—
251#106—44no—
252C4⋊C8—24no—
253C4⋊C8—24no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—42no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259#97—24no—
260#97—41no—
261C4×C8—22no—
262C4×C8—24no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8C2×Q818yes—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95C2×D411yescharacteristic
280#95C2×D412yes—
281#95C2×D411yescharacteristic
282#95C2×D412yes—
283#95C2×D412yes—
284#95C2×D412yes—
285#95C2×D412yes—
286C2×Q8—216no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—44no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—82no—
298C2×Q8—84no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2511yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8C22×Q814yes—
326C2×C8C22×Q814yes—
327C2×C8H(5,𝔽2)12yes—
328C4⋊C4—216no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4C22×Q818yes—
355C4⋊C4C22×Q818yes—
356C42—21no—
357C42—22no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—22no—
362C42—21no—
363C42—21no—
364C42—44no—
365C42—84no—
366C42—84no—
367C42—84no—
368C42—82no—
369C42C22×Q814yes—
370C422−1+414yes—
371C23C23×D411yescharacteristic
372Q8—44no—
373Q8—44no—
374Q8—816no—
375Q8—164no—
376Q8—164no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4D4×Q814yes—
413C2×C4D4×Q814yes—
414C2×C4#57612yes—
415C2×C4#57812yes—
416C2×C4C23×Q812yes—
417C2×C4C2×H(5,𝔽2)11yescharacteristic
418C2×C4C2×2−1+412yes—
419C2×C4#58511yescharacteristic
420C8—24no—
421C8—41no—
422C8—42no—
423C8—41no—
424C8—41no—
425C8Q8214yes—
426C22C2×D4×Q812yes—
427C22#349611yescharacteristic
428C22C2×#57611yescharacteristic
429C22C2×#57811yescharacteristic
430C22#358412yes—
431C4—22no—
432C4—22no—
433C4—24no—
434C4—24no—
435C4—42no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—48no—
442C4—48no—
443C4—84no—
444C4—84no—
445C4C2×Q8211yescharacteristic
446C4#350212yes—
447C4#355011yescharacteristic
448C2#5983211yescharacteristic
449C2#6137012yes—
450C2#6137212yes—
451C2#6227211yescharacteristic
452C2#6227511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2399 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
91. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#453642
#326942
#326912
#292932
#288122
#286792
#286762
#286752
#264552
#264542
#231582
#231572
#217252
#217152