Finite Group Databasebeta

← #6437272

Subgroups — #6437272

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34625C212yesmaximal
2#30315C214yesmaximal, complemented
3#30313C214yesmaximal, complemented
4#30301C212yesmaximal
5#30300C212yesmaximal
6#30299C214yesmaximal
7#23838C214yesmaximal, complemented
8#23836C212yesmaximal, complemented
9#23835C212yesmaximal, complemented
10#20819C212yesmaximal
11#20818C211yesmaximal, characteristic
12#20817C211yesmaximal, characteristic
13#20366C211yesmaximal, characteristic
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2667—22no—
17#2666—24no—
18#2665—22no—
19#2660C2214yes—
20#2647C2214yes—
21#2629C2214yes—
22#2628C2214yes—
23#2402—22no—
24#2399—22no—
25#2396—24no—
26C4×C4⋊Q8C2214yes—
27C4×#125C2214yes—
28#2285C2212yes—
29#2283C2212yes—
30C4×#117C2211yescharacteristic
31#2085C2214yes—
32#2085C2214yes—
33#2085C2214yes—
34#2085C2214yes—
35#2084C2214yes—
36#2083C2214yes—
37#2083C2214yes—
38#2083C2214yes—
39#2059C2214yes—
40#2059C2214yes—
41#2059C2214yes—
42#2059C2214yes—
43#2058C2214yes—
44#2057C2214yes—
45#2057C2214yes—
46#2057C2214yes—
47#1847C2214yes—
48#1846C2212yes—
49#1845C2212yes—
50#1836C2214yes—
51#1835C2212yes—
52#1834C2212yes—
53#1809C2214yes—
54#1809C2214yes—
55#1808C2212yes—
56#1808C2211yescharacteristic
57#1808C2211yescharacteristic
58#1808C2212yes—
59#1807C2212yes—
60C4×#107C2211yescharacteristic
61C4×#107C2212yes—
62C4×#107C2214yes—
63C4×#107C2211yescharacteristic
64C4×#107C2212yes—
65C4×#106C2214yes—
66C4×#106C2212yes—
67#1746C2212yes—
68#1746C2212yes—
69#1744C2212yes—
70#1744C2212yes—
71#1729C2212yes—
72#1729C2212yes—
73C22×M16C2312yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×C4⋊Q8C2314yes—
77C2×#125C2314yes—
78C2×C4×Q8—24no—
79C2×#117—22no—
80C2×#117C2312yes—
81C22×C42C2311yescharacteristic
82C2×#107C2312yes—
83C2×#107C2312yes—
84C2×#107C2314yes—
85C2×#107C2314yes—
86C2×#107C2314yes—
87C2×#107C2312yes—
88C2×#107C2312yes—
89C2×#106C2314yes—
90C2×#106C2314yes—
91C2×#106C2314yes—
92C4×M16Q818yescomplemented
93C2×#97C2312yes—
94C2×C4×C8Q818yescomplemented
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2312yes—
100#400—24no—
101#400—24no—
102#400—24no—
103#399—24no—
104#398—24no—
105#398—24no—
106#398—24no—
107#398—24no—
108#398—24no—
109#398—24no—
110#396—24no—
111#396—24no—
112#395—24no—
113#395—24no—
114#395—24no—
115#393—24no—
116#391—24no—
117#389—24no—
118#389C2314yes—
119#389C2314yes—
120#389C2314yes—
121#389C2314yes—
122#387—24no—
123#387—24no—
124#384C2314yes—
125#384C2314yes—
126#383—22no—
127#382—22no—
128#382C2314yes—
129#382C2314yes—
130#381—22no—
131#380—22no—
132#380C2314yes—
133#380C2312yes—
134#380C2312yes—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D412yes—
142C4×C4⋊C4C2312yes—
143C4×C4⋊C4C2314yes—
144C4×C4⋊C4C2312yes—
145C4×C22⋊C4—22no—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4D412yes—
148#376—22no—
149#376C2312yes—
150C43C2312yes—
151#340C2312yes—
152#340C2312yes—
153#340C2314yes—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2312yes—
159#340C2312yes—
160#340C2314yes—
161#336C2314yes—
162#336C2314yes—
163#336C2314yes—
164#336C2314yes—
165C22×Q8—24no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—42no—
170C2×M16—42no—
171C2×M16C2×Q818yes—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×Q814yes—
175C22×C8C2×Q814yes—
176C22×C8C2411yescharacteristic
177C22×C8C2411yescharacteristic
178C22×C8C2411yescharacteristic
179C22×C8C2411yescharacteristic
180C22×C8C2414yes—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183#125—44no—
184#125—44no—
185C4×Q8—88no—
186#117—42no—
187#117—42no—
188#117—84no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4C2×D414yes—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D414yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q814yes—
219C2×C42C2×Q814yes—
220C2×C42C2414yes—
221C2×C42C2411yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—44no—
225#107—42no—
226#107—44no—
227#107—42no—
228#107—44no—
229#107—42no—
230#106—44no—
231#106—44no—
232#106—44no—
233#97C2×Q818yes—
234C4×C8C2×Q814yes—
235C4×C8C2×Q814yes—
236C4×C8C2×Q814yes—
237C4×C8C2×Q814yes—
238C4×C8C2×Q818yes—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—22no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95C2×D412yes—
254#95C2×D414yes—
255#95C2×D412yes—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4C4⋊Q814yes—
270C22×C4C4⋊Q814yes—
271C22×C4C22×D414yes—
272C22×C4C22×D414yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q812yes—
279C22×C4C22×Q812yes—
280C22×C4C2511yescharacteristic
281M16—48no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8C22×Q812yes—
291C2×C8C22×Q812yes—
292C2×C8C22×Q812yes—
293C2×C8C22×Q812yes—
294C2×C8C22×Q818yes—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C22⋊C4—44no—
317C22⋊C4—84no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—44no—
325C42—48no—
326C42—44no—
327C42—44no—
328C42C4⋊Q814yes—
329C42C4⋊Q814yes—
330C42C22×Q812yes—
331C42C22×Q812yes—
332C42C22×Q812yes—
333C42C22×Q812yes—
334C23—22no—
335C23—22no—
336C23—21no—
337C23—21no—
338C23C2×C4⋊Q812yes—
339C23C23×D411yescharacteristic
340Q8—816no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4C2×C4⋊Q812yes—
370C2×C4C2×C4⋊Q812yes—
371C2×C4C2×C4⋊Q812yes—
372C2×C4C2×C4⋊Q812yes—
373C2×C4C2×C4⋊Q812yes—
374C2×C4#57612yes—
375C2×C4#57812yes—
376C2×C4C23×Q811yescharacteristic
377C2×C4C23×Q811yescharacteristic
378C2×C4#58512yes—
379C2×C4#58512yes—
380C8—22no—
381C8—22no—
382C8—22no—
383C8—22no—
384C8—28no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22C22×C4⋊Q811yescharacteristic
391C22#344912yes—
392C22C2×#57611yescharacteristic
393C22C2×#57811yescharacteristic
394C22#358412yes—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—44no—
400C4—44no—
401C4—84no—
402C4—84no—
403C4—84no—
404C4—84no—
405C4#316414yes—
406C4#345811yescharacteristic
407C4#345811yescharacteristic
408C4#345812yes—
409C2—42no—
410C2#3464211yescharacteristic
411C2C2×#316411yescharacteristic
412C2#6066011yescharacteristic
413C2#6095512yes—
414C2#6095712yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #30315C2semidirect
3. #30313C2semidirect
7. #23838C2semidirect
8. #23836C2semidirect
9. #23835C2semidirect
92. C4×M16Q8semidirect
94. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#346252
#303152
#303132
#303012
#303002
#302992
#238382
#238362
#238352
#208192
#208182
#208172
#203662