Finite Group Databasebeta

← #193369

Subgroups — #193369

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic
4C2×#1542C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7884C211yesmaximal, characteristic
9#7879C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962C2214yes—
16#2961—22no—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#1950—22no—
25#1950C2214yes—
26#1810—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1568—22no—
31#1542C414yes—
32#1542C2212yes—
33#1542C2212yes—
34C2×#333C412yes—
35C2×#332C412yes—
36#1497—22no—
37#1493C2214yes—
38#1479—22no—
39#1278C2212yes—
40#1278C2212yes—
41#1277C2214yes—
42C23×C8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
49#533Q812yes—
50#533C2312yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8Q812yes—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C22×C42C2311yescharacteristic
60#446—28no—
61#446—28no—
62#446—28no—
63#446—24no—
64C8×Q8—24no—
65#433—28no—
66#432—24no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432D412yes—
74#432D412yes—
75#431—24no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431C2×C412yes—
87#431C2311yescharacteristic
88#431C2311yescharacteristic
89C2×#108—24no—
90#424—24no—
91#423D412yes—
92#423C2312yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C412yes—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8C2×C412yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102#407—21no—
103#407D412yes—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107#405—24no—
108C4×M16—21no—
109C4×M16C2312yes—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113#341—24no—
114#335—24no—
115#334—24no—
116#333C2×C414yes—
117#333C2×C414yes—
118#332C2×C414yes—
119#332C2×C414yes—
120Q8⋊C8—28no—
121C22×Q8—21no—
122C22×Q8C4⋊C412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16C2×D414yes—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—82no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C4⋊C412yes—
140C4⋊Q8C4⋊C412yes—
141C4⋊Q8C22×C412yes—
142C4⋊Q8C22×C412yes—
143#125—22no—
144#125—22no—
145#125—22no—
146#125—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8C4⋊C414yes—
154C22⋊Q8C2×Q814yes—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8C4⋊C414yes—
162C4×Q8C2×D414yes—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C4⋊C412yes—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×Q811yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#108—84no—
188C4⋊C8—28no—
189C4⋊C8—28no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8C22×C414yes—
202C4⋊C8C22×C412yes—
203C4⋊C8C22×C412yes—
204C4⋊C8C2×D414yes—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—28no—
217#97—22no—
218#97—22no—
219#97—22no—
220#97—41no—
221C4×C8—28no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—41no—
229C24C22⋊Q811yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8C2×C4⋊C414yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260M16—24no—
261M16—24no—
262M16—42no—
263M16—42no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—44no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—82no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4C2×C4⋊C414yes—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C42—24no—
293C42—24no—
294C42C2×C4⋊C411yescharacteristic
295C42C2×C4⋊C411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12011yescharacteristic
301C42#12011yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42C23×C411yescharacteristic
307C42C22×D411yescharacteristic
308C42C22×D411yescharacteristic
309C42C2×#4111yescharacteristic
310C23—22no—
311C23—22no—
312C23—41no—
313C23#38511yescharacteristic
314C23#38511yescharacteristic
315C23C2×C22⋊Q811yescharacteristic
316Q8—44no—
317Q8—44no—
318Q8—82no—
319Q8—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4#38511yescharacteristic
344C2×C4#38511yescharacteristic
345C2×C4#38511yescharacteristic
346C2×C4#38511yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4C22×C4⋊C411yescharacteristic
350C2×C4C2×C4×D411yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×#12011yescharacteristic
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4C2×#12311yescharacteristic
355C2×C4#57712yes—
356C2×C4#57812yes—
357C8—44no—
358C8—42no—
359C8—42no—
360C8—44no—
361C8—44no—
362C8—82no—
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—44no—
372C22#185812yes—
373C22C2×#38511yescharacteristic
374C22#308212yes—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—44no—
385C4—44no—
386C4#294911yescharacteristic
387C4#295011yescharacteristic
388C4#295712yes—
389C4#301312yes—
390C2—22no—
391C2—22no—
392C2—41no—
393C2C2×#185811yescharacteristic
394C2#2088011yescharacteristic
395C2#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
#219692
#178162
C2×#15422
#174792
#89332
#89312
#78842
#78792