Finite Group Databasebeta

← #192290

Subgroups — #192290

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21301C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17817C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#8865C218yesmaximal, complemented
7#8088C211yesmaximal, characteristic
8C2×#1302C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2987—22no—
11#2985—22no—
12#2984—22no—
13#2983C2214yes—
14#2962—21no—
15#2962—21no—
16#2961—21no—
17#2961—21no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#1985—22no—
26#1985C2214yes—
27#1729—22no—
28#1574—22no—
29#1573C2214yes—
30#1573C2214yes—
31#1571—21no—
32#1570—21no—
33#1569—22no—
34#1567—21no—
35#1566—21no—
36#1472C414yes—
37#1470C414yes—
38#1461C2214yes—
39#1302C2214yes—
40#1302C2214yes—
41C22×M16—21no—
42C23×C8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51C2×C4⋊Q8Q811yescharacteristic
52C2×C4⋊Q8Q811yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8Q812yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8C2312yes—
58C2×#117D412yes—
59C22×C42C2311yescharacteristic
60#446—24no—
61#446—24no—
62#446—22no—
63#446—22no—
64#446—24no—
65#446—24no—
66#446—24no—
67#446—24no—
68C8×Q8—22no—
69C8×Q8—22no—
70#433—24no—
71#433—24no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—24no—
76#432—21no—
77#432C2×C412yes—
78#432D412yes—
79#432C2312yes—
80#432C2312yes—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—24no—
85#431—21no—
86#431C2×C412yes—
87#431D412yes—
88#431C2312yes—
89#431C2312yes—
90#424—22no—
91#424—22no—
92#423—21no—
93#423—21no—
94#423—21no—
95#423—21no—
96#423—21no—
97#423D411yescharacteristic
98#423D411yescharacteristic
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2312yes—
104#407—21no—
105#407C2×C412yes—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C2×C412yes—
108#405—22no—
109#405—22no—
110C4×M16—21no—
111C4×M16—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97C2312yes—
119C2×C4×C8—21no—
120#336—28no—
121#335—24no—
122#335—24no—
123#334—24no—
124#334—24no—
125#324C2×C4116yes—
126C22×Q8C4⋊C412yes—
127C22×Q8C2×Q812yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q811yescharacteristic
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—44no—
135C2×M16—41no—
136C2×M16—41no—
137C22×C8—24no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—41no—
145C22×C8—41no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
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⋊Q8—22no—
165C22⋊Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4C4⋊C412yes—
179C2×C4⋊C4C4⋊C412yes—
180C2×C4⋊C4C4⋊C412yes—
181C2×C4⋊C4C4⋊C414yes—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C22⋊C4C4⋊C414yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D414yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8C2×D412yes—
215C4⋊C8C2×D412yes—
216C4⋊C8C2×D412yes—
217C4⋊C8C2×D412yes—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8C22×C414yes—
225C22⋊C8C22×C414yes—
226#97—24no—
227#97—24no—
228#97—22no—
229#97—22no—
230#97—22no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97—41no—
235#97—41no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C24C2×C4⋊C411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—82no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4C2×C4⋊C414yes—
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C414yes—
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×Q811yescharacteristic
273M16—42no—
274M16—42no—
275M16—82no—
276M16—82no—
277M16—82no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—82no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C42—24no—
309C42—24no—
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42#12011yescharacteristic
315C42#12011yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42#12311yescharacteristic
321C42#12311yescharacteristic
322C42C22×D411yescharacteristic
323C42C22×D411yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—22no—
327C23—22no—
328C23—41no—
329C23#38511yescharacteristic
330C23#38511yescharacteristic
331C23C22×C4⋊C411yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—82no—
335Q8—82no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—41no—
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#38511yescharacteristic
360C2×C4#38511yescharacteristic
361C2×C4#38511yescharacteristic
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#45412yes—
366C2×C4#45612yes—
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×#12011yescharacteristic
370C2×C4C2×C22⋊Q811yescharacteristic
371C2×C4C2×C22⋊Q811yescharacteristic
372C2×C4C2×#12311yescharacteristic
373C8—44no—
374C8—44no—
375C8—44no—
376C8—42no—
377C8—42no—
378C8—42no—
379C8—42no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—44no—
389C22#185712yes—
390C22C2×#38511yescharacteristic
391C22#291612yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—44no—
404C4#185712yes—
405C4C2×#45411yescharacteristic
406C4C2×#45611yescharacteristic
407C2—22no—
408C2—22no—
409C2—41no—
410C2#2085211yescharacteristic
411C2C2×#185711yescharacteristic
412C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ 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. #8865C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#213012
#178222
#178172
#173852
#88652
#80882
C2×#13022