Finite Group Databasebeta

← #368980

Subgroups — #368980

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#35335C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4C2×#1998C211yesmaximal, characteristic
5C2×#1998C211yesmaximal, characteristic
6#11206C214yesmaximal
7#10916C214yesmaximal
8#10856C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×#125C2211yescharacteristic
12Q8×M16—24no—
13#2962—24no—
14#2961—22no—
15#2960—22no—
16#2905C2212yes—
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19#2873C2212yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2859—22no—
23#2660—22no—
24#2584—24no—
25#2583C418yes—
26#2317C2214yes—
27C4×#117C2211yescharacteristic
28#1998—22no—
29#1998—22no—
30#1998—24no—
31#1998—22no—
32#1997—22no—
33C2×#344C2214yes—
34C2×#344C2212yes—
35C2×#344C2212yes—
36#1742C2214yes—
37C2×#343C2212yes—
38C2×#343C2214yes—
39C2×#343C2212yes—
40C22×M16C2312yes—
41C22×M16C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45#533—21no—
46#533—21no—
47#533—22no—
48#533—21no—
49#533—21no—
50#533—22no—
51C2×C4⋊Q8D412yes—
52C2×#125—22no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—22no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—22no—
61C2×#125D412yes—
62C2×C22⋊Q8C2×C414yes—
63C2×C4×Q8D414yes—
64C2×#117C2312yes—
65C2×#117C2312yes—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2312yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#446—216no—
71C8×Q8—28no—
72#424—24no—
73#424—24no—
74#423—22no—
75#423—24no—
76#423—21no—
77#423—22no—
78#423—21no—
79#423D412yes—
80#423D412yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83#405—24no—
84#405—24no—
85C4×M16—22no—
86C4×M16—24no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16D414yes—
92C2×#97—21no—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#398—28no—
97#395—28no—
98#389—216no—
99#389—28no—
100#385C2×C4116yes—
101#384—24no—
102#382—24no—
103C4×C4⋊C4—22no—
104C4×C22⋊C4C2×C414yes—
105#376—21no—
106C43—21no—
107#344—24no—
108#344—28no—
109#344Q818yescomplemented
110#343—28no—
111#343—24no—
112#343—24no—
113#343—28no—
114C22×Q8C22⋊C414yes—
115C23×C4C22×C414yes—
116C23×C4C22×C412yes—
117C23×C4C2411yescharacteristic
118C2×M16—21no—
119C2×M16—22no—
120C2×M16—21no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—24no—
126C2×M16—22no—
127C2×M16—24no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16C2×D412yes—
136C2×M16C2×D414yes—
137C2×M16C2×D412yes—
138C2×M16C2×Q814yes—
139C2×M16C2×Q814yes—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149#125—22no—
150#125—22no—
151#125—24no—
152#125—42no—
153#125—42no—
154#125—41no—
155#125—41no—
156#125—41no—
157#125—41no—
158#125—42no—
159#125—42no—
160#125—42no—
161#125—42no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117C2×Q814yes—
177#117#4114yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4C22⋊C418yes—
193C2×C4⋊C4C22⋊C414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4C22×C414yes—
196C2×C22⋊C4C22×C414yes—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42C2×D414yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4112yes—
209C4⋊C8—216no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214#97—28no—
215#97—24no—
216#97—41no—
217#97—41no—
218C4×C8—28no—
219C4×C8—24no—
220C4×C8—41no—
221C4×C8—41no—
222#95—24no—
223#95—216no—
224C24C23×C411yescharacteristic
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—84no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—82no—
233C22×C4—84no—
234C22×C4C2×C22⋊C418yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C4×D412yes—
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4C4×D412yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×Q812yes—
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4112yes—
249C22×C42−1+412yes—
250C22×C42−1+412yes—
251M16—22no—
252M16—24no—
253M16—22no—
254M16—22no—
255M16—28no—
256M16—24no—
257M16—24no—
258M16—42no—
259M16—42no—
260M16—44no—
261M16—42no—
262M16—42no—
263M16—42no—
264M16—44no—
265M16C4⋊Q814yes—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—816no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4C4×Q818yes—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—84no—
304C42#12012yes—
305C42#12012yes—
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12112yes—
311C42#12112yes—
312C42#12312yes—
313C42#12312yes—
314C23—42no—
315C23C22×C22⋊C411yescharacteristic
316C23C2×C4×D411yescharacteristic
317C23C2×C4×D411yescharacteristic
318C23C2×C4×Q812yes—
319C23#51912yes—
320Q8—48no—
321Q8—84no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4#38612yes—
337C2×C4#38612yes—
338C2×C4#38612yes—
339C2×C4#38612yes—
340C2×C4C2×#12011yescharacteristic
341C2×C4C2×#12111yescharacteristic
342C2×C4C2×#12111yescharacteristic
343C2×C4C2×#12311yescharacteristic
344C2×C4C2×C4⋊Q811yescharacteristic
345C2×C4#53211yescharacteristic
346C2×C4#53312yes—
347C2×C4#53611yescharacteristic
348C2×C4#53611yescharacteristic
349C2×C4#54112yes—
350C2×C4#54712yes—
351C2×C4#54812yes—
352C2×C4#54812yes—
353C2×C4D4×Q812yes—
354C2×C4D4×Q812yes—
355C2×C4D4×Q812yes—
356C2×C4#55612yes—
357C2×C4#55712yes—
358C8—24no—
359C8—44no—
360C8—44no—
361C8—42no—
362C8—82no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22C2×#38611yescharacteristic
369C22C4×C4⋊Q811yescharacteristic
370C22#232711yescharacteristic
371C22Q8×C22⋊C412yes—
372C22#236512yes—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—48no—
378C4—48no—
379C4—84no—
380C4#242912yes—
381C4#243211yescharacteristic
382C4#243411yescharacteristic
383C4#244612yes—
384C4#254511yescharacteristic
385C4#254611yescharacteristic
386C2—22no—
387C2—21no—
388C2—21no—
389C2—22no—
390C2#1402411yescharacteristic
391C2#2101712yes—
subgroupG/H
center, socleC22C2×#386
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
109. #344Q8semidirect

Maximal subgroups

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

subgroupindex
#601922
#353352
#213372
C2×#19982
#112062
#109162
#108562
C4×#3442