Finite Group Databasebeta

← #193377

Subgroups — #193377

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21970C211yesmaximal, characteristic
3#17803C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7884C211yesmaximal, characteristic
9#7880C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986—22no—
12#2985—22no—
13#2984—22no—
14#2983—21no—
15#2982—21no—
16#2962C2214yes—
17#2960—22no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#1950—22no—
26#1950C2214yes—
27#1811—22no—
28#1572—21no—
29#1572—21no—
30#1571C2212yes—
31#1570C2212yes—
32#1564—22no—
33#1550C414yes—
34#1549C412yes—
35#1548C412yes—
36#1542C2212yes—
37#1542C2212yes—
38#1497—22no—
39#1493C2214yes—
40#1479—22no—
41#1278C2212yes—
42#1278C2212yes—
43#1277C2214yes—
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—28no—
65#446—24no—
66C8×Q8—24no—
67C8×Q8—24no—
68C8×Q8—24no—
69#433—28no—
70#432—24no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
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#431—21no—
87#431—21no—
88#431C2×C412yes—
89#431D412yes—
90#431D411yescharacteristic
91#431D411yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94C2×#108—24no—
95#424—24no—
96#423—21no—
97#423—21no—
98#423C2×C412yes—
99#423C2×C412yes—
100#423D412yes—
101#423C2312yes—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C412yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#407—21no—
108#407D412yes—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112#405—24no—
113C4×M16—21no—
114C4×M16C2312yes—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2312yes—
118#341—24no—
119#335—22no—
120#335—22no—
121#334—22no—
122#334—22no—
123#333C2×C414yes—
124#333C2×C414yes—
125#332C2×C414yes—
126#332C2×C414yes—
127Q8⋊C8—28no—
128C22×Q8—21no—
129C22×Q8C4⋊C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q811yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—82no—
137C2×M16C2×D414yes—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8C4⋊C412yes—
147C4⋊Q8C4⋊C412yes—
148C4⋊Q8C22×C412yes—
149C4⋊Q8C22×C412yes—
150#125—22no—
151#125—22no—
152#125—22no—
153#125—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8C4⋊C414yes—
161C22⋊Q8C2×Q814yes—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8C4⋊C414yes—
169C4×Q8C2×D414yes—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C4⋊C412yes—
180C2×C22⋊C4—22no—
181C2×C42—22no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42C2411yescharacteristic
194#108—84no—
195C4⋊C8—28no—
196C4⋊C8—28no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8C22×C412yes—
209C4⋊C8C22×C412yes—
210C4⋊C8C22×C414yes—
211C4⋊C8C2×D412yes—
212C4⋊C8C2×D412yes—
213C22⋊C8—22no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225#97—28no—
226#97—22no—
227#97—41no—
228C4×C8—28no—
229C4×C8—22no—
230C4×C8—21no—
231C4×C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—41no—
239C24C22⋊Q811yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8C2×C4⋊C414yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4111yescharacteristic
270M16—24no—
271M16—24no—
272M16—42no—
273M16—42no—
274M16—82no—
275M16—82no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4C2×C4⋊C414yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C42—24no—
304C42—24no—
305C42C2×C4⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12011yescharacteristic
312C42#12011yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42C23×C411yescharacteristic
318C42C22×D411yescharacteristic
319C42C22×D411yescharacteristic
320C42C2×#4111yescharacteristic
321C23—22no—
322C23—41no—
323C23—41no—
324C23#38511yescharacteristic
325C23#38511yescharacteristic
326C23C2×C22⋊Q811yescharacteristic
327Q8—44no—
328Q8—44no—
329Q8—44no—
330Q8—44no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—82no—
354C2×C4#38511yescharacteristic
355C2×C4#38511yescharacteristic
356C2×C4#38511yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38511yescharacteristic
360C2×C4C22×C4⋊C411yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×#12011yescharacteristic
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×#12311yescharacteristic
366C2×C4#57712yes—
367C2×C4#57812yes—
368C8—44no—
369C8—42no—
370C8—42no—
371C8—42no—
372C8—42no—
373C8—44no—
374C8—82no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—82no—
385C22C2×#38511yescharacteristic
386C22#308212yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—44no—
397C4#185812yes—
398C4#294911yescharacteristic
399C4#295011yescharacteristic
400C4#295512yes—
401C4#301312yes—
402C2—22no—
403C2—41no—
404C2—41no—
405C2C2×#185811yescharacteristic
406C2#2088011yescharacteristic
407C2#2088411yescharacteristic
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
#602502
#219702
#178032
#177082
#174792
#89332
#89312
#78842
#78802