Finite Group Databasebeta

← #238163

Subgroups — #238163

Order 512.

896 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#37642C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5#10353C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8038C211yesmaximal, characteristic
8#7896C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2844—22no—
15#2628—22no—
16#2628—22no—
17#2628—22no—
18#2571—21no—
19#2571—21no—
20#2571C2212yes—
21#2551—21no—
22#2551—21no—
23#2551C2212yes—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1993—22no—
28#1990—21no—
29#1990—21no—
30#1990C2212yes—
31#1990C2212yes—
32#1988C2214yes—
33#1981C2214yes—
34#1956—21no—
35#1956—21no—
36#1823—22no—
37#1787—22no—
38#1736—22no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C22×M16—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yescomplemented
55C2×C4⋊Q8C2312yes—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117D412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—24no—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—22no—
73C4×M16—22no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2312yes—
78#400—24no—
79#400—24no—
80#400—24no—
81#399—24no—
82#395—24no—
83#395—24no—
84#395—24no—
85#393—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#382—22no—
99#382—22no—
100#382—22no—
101#382—21no—
102#382—21no—
103#382D412yes—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380D412yes—
109#380D412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2312yes—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—82no—
126C22×Q8—82no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×M16—24no—
132C2×M16—42no—
133C2×M16—42no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—84no—
147C4⋊Q8—84no—
148C4⋊Q8—84no—
149C4⋊Q8C2×Q814yes—
150#117—44no—
151#117—82no—
152#117—82no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q812yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42C2411yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200#103—84no—
201#103—84no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204#97—42no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—42no—
208C4×C8—42no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219C24#12011yescharacteristic
220C2×Q8—24no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—88no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—88no—
229C2×Q8—84no—
230C2×Q8—84no—
231C2×Q8—84no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4C22⋊Q812yes—
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12412yes—
255C22×C4#12412yes—
256C22×C4#12412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260M16—48no—
261C2×C8—24no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C42—44no—
301C42—44no—
302C42—44no—
303C42C22⋊Q812yes—
304C42C22⋊Q812yes—
305C42C22⋊Q812yes—
306C42#12412yes—
307C42#12412yes—
308C42#12412yes—
309C42C22×Q811yescharacteristic
310C42C2×#4111yescharacteristic
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#39312yes—
320C23C2×#12011yescharacteristic
321Q8—48no—
322Q8—88no—
323Q8—84no—
324Q8—168no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—88no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#39312yes—
360C2×C4#39312yes—
361C2×C4#39312yes—
362C2×C4#45512yes—
363C2×C4#45612yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×#12411yescharacteristic
368C2×C4C2×#12411yescharacteristic
369C2×C4C2×#12411yescharacteristic
370C2×C4#57612yes—
371C2×C4#57712yes—
372C2×C4#57712yes—
373C2×C4#57812yes—
374C8—44no—
375C8—82no—
376C8—82no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22C2×#39311yescharacteristic
385C22#300811yescharacteristic
386C22#300911yescharacteristic
387C22#301011yescharacteristic
388C22#301111yescharacteristic
389C22C2×#45511yescharacteristic
390C22C2×#45611yescharacteristic
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—82no—
400C4—82no—
401C4—84no—
402C4#202814yes—
403C4#308112yes—
404C4#308212yes—
405C4#313612yes—
406C4#313712yes—
407C2—42no—
408C2#1872211yescharacteristic
409C2#1872511yescharacteristic
410C2#1873511yescharacteristic
411C2#1873611yescharacteristic
412C2#2345611yescharacteristic
413C2#2345711yescharacteristic
414C2C2×#202811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10353C2semidirect
6. #10350C2semidirect
54. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#218332
#218042
#213242
#103532
#103502
#80382
#78962