Finite Group Databasebeta

← #193360

Subgroups — #193360

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17718C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7886C211yesmaximal, characteristic
9#7878C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962C2212yes—
16#2962C2212yes—
17#2961—21no—
18#2961—21no—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26#1950C2214yes—
27#1855—22no—
28#1848—21no—
29#1837—21no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1569—22no—
34#1554C414yes—
35#1548C412yes—
36#1546C412yes—
37#1542C2214yes—
38#1493C2214yes—
39#1487—22no—
40#1481—22no—
41#1278C2214yes—
42#1277C2212yes—
43#1277C2212yes—
44C22×M16—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533D412yes—
51#533Q812yes—
52#533C2312yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8Q812yes—
58C2×C4×Q8—21no—
59C2×C4×Q8D412yes—
60C2×#117—21no—
61C22×C42C2311yescharacteristic
62#446—24no—
63#446—24no—
64#446—24no—
65#446—24no—
66#446—24no—
67#446—24no—
68#446—22no—
69#446—22no—
70C8×Q8—22no—
71C8×Q8—22no—
72#433—24no—
73#433—24no—
74#432—24no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432C2×C412yes—
81#432D412yes—
82#432D412yes—
83#431—24no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431C2×C412yes—
94#431C2×C412yes—
95#431C2312yes—
96C2×#108—24no—
97#424—22no—
98#424—22no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423C2×C412yes—
103#423D411yescharacteristic
104#423D411yescharacteristic
105#423C2312yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112#407—21no—
113#407D412yes—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117#405—22no—
118#405—22no—
119C4×M16—21no—
120C4×M16C2312yes—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2312yes—
124#341—24no—
125#335—24no—
126#334—24no—
127#333C2×C418yes—
128#332C2×C418yes—
129Q8⋊C8—28no—
130C22×Q8—21no—
131C22×Q8C4⋊C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×Q811yescharacteristic
134C23×C4#4111yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—41no—
138C2×M16—82no—
139C2×M16C2×D412yes—
140C2×M16C2×D412yes—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—41no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C4⋊C412yes—
151C4⋊Q8C4⋊C412yes—
152C4⋊Q8C22×C412yes—
153C4⋊Q8C22×C412yes—
154#125—22no—
155#125—22no—
156#125—22no—
157#125—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8C4⋊C414yes—
165C22⋊Q8C2×Q814yes—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8C4⋊C414yes—
173C4×Q8C2×D414yes—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C4⋊C412yes—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198#108—84no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8C22×C414yes—
218C4⋊C8C22×C414yes—
219C4⋊C8C2×D412yes—
220C4⋊C8C2×D412yes—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231#97—24no—
232#97—24no—
233#97—22no—
234#97—22no—
235#97—22no—
236#97—22no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—22no—
245C24C22⋊Q811yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8C2×C4⋊C414yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×Q811yescharacteristic
275C22×C4C2×#4111yescharacteristic
276M16—24no—
277M16—24no—
278M16—44no—
279M16—82no—
280M16—82no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—82no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4C2×C4⋊C414yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C42—24no—
310C42—24no—
311C42C2×C4⋊C411yescharacteristic
312C42C2×C4⋊C411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42#12011yescharacteristic
318C42#12011yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42#12311yescharacteristic
322C42#12311yescharacteristic
323C42C23×C411yescharacteristic
324C42C22×D411yescharacteristic
325C42C22×D411yescharacteristic
326C42C2×#4111yescharacteristic
327C23—22no—
328C23—41no—
329C23—41no—
330C23#38511yescharacteristic
331C23#38511yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333Q8—44no—
334Q8—44no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—82no—
360C2×C4#38511yescharacteristic
361C2×C4#38511yescharacteristic
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4C22×C4⋊C411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×#12011yescharacteristic
370C2×C4C2×C22⋊Q811yescharacteristic
371C2×C4C2×#12311yescharacteristic
372C2×C4#57712yes—
373C2×C4#57812yes—
374C8—44no—
375C8—44no—
376C8—42no—
377C8—42no—
378C8—42no—
379C8—42no—
380C8—82no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—82no—
391C22C2×#38511yescharacteristic
392C22#308212yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—44no—
404C4#294712yes—
405C4#295712yes—
406C4#301011yescharacteristic
407C4#301111yescharacteristic
408C2—22no—
409C2—41no—
410C2—41no—
411C2#2086011yescharacteristic
412C2#2087711yescharacteristic
413C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8933C2semidirect
7. #8931C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#219932
#178222
#177182
#174862
#89332
#89312
#78862
#78782