Finite Group Databasebeta

← #193406

Subgroups — #193406

Order 512.

793 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic
3#17823C211yesmaximal, characteristic
4#17808C211yesmaximal, characteristic
5#17739C211yesmaximal, characteristic
6#8936C214yesmaximal, complemented
7#8936C214yesmaximal, complemented
8#7870C211yesmaximal, characteristic
9#7868C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986C2214yes—
12#2985—22no—
13#2984—22no—
14#2983—21no—
15#2982—21no—
16#2962—22no—
17#2960—22no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1994C2214yes—
26#1986—22no—
27#1826—22no—
28#1574C2212yes—
29#1574C2212yes—
30#1574C2214yes—
31#1572—22no—
32#1572—21no—
33#1572—21no—
34#1569—22no—
35#1565—22no—
36#1561C414yescomplemented
37#1556C412yescomplemented
38#1556C412yescomplemented
39#1544C2212yes—
40#1544C2212yes—
41#1276C2212yes—
42#1276C2212yes—
43#1276C2214yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533Q812yes—
51#533C2312yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125Q812yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×#117—21no—
60C2×#117D412yes—
61C22×C42C2311yescharacteristic
62#446—24no—
63#446—28no—
64#446—24no—
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#432—21no—
76#432C2×C412yes—
77#432C2×C412yes—
78#432D412yes—
79#431—24no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431D411yescharacteristic
88#431D411yescharacteristic
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2312yes—
92#431C2312yes—
93#431C2312yes—
94#424—24no—
95#423C2×C412yes—
96#423C2×C412yes—
97#423D412yes—
98#423C2312yes—
99C2×C4⋊C8D412yes—
100#407—21no—
101#407—21no—
102#407—21no—
103#407—21no—
104C2×C22⋊C8—21no—
105#405—24no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16—21no—
110C2×C4×C8—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115#376—21no—
116C43C2312yes—
117#339—28no—
118#335—22no—
119#335—22no—
120#335—24no—
121#334—22no—
122#334—22no—
123#334—24no—
124#332C2×C414yes—
125#332C2×C414yes—
126#332C2×C414yes—
127#332C2×C414yes—
128C22×Q8—21no—
129C22×Q8—21no—
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C22×C8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C4⋊C412yes—
143C4⋊Q8C4⋊C412yes—
144C4⋊Q8C22×C412yes—
145C4⋊Q8C22×C412yes—
146#125—22no—
147#125—22no—
148#125C4⋊C414yes—
149#125C2×Q814yes—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166#117—22no—
167#117—22no—
168#117—44no—
169#117C4⋊C414yes—
170#117C2×D414yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4C4⋊C412yes—
177C2×C4⋊C4C4⋊C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196C4⋊C8—28no—
197C4⋊C8—28no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8C22×C412yes—
205C4⋊C8C22×C412yes—
206C4⋊C8C22×C414yes—
207C4⋊C8C2×D412yes—
208C4⋊C8C2×D412yes—
209C4⋊C8C2×D414yes—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—22no—
220C22⋊C8—41no—
221C22⋊C8—41no—
222#97—28no—
223#97—22no—
224#97—22no—
225#97—22no—
226#97—41no—
227C4×C8—28no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—41no—
236#95—41no—
237#95—41no—
238C24#12311yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—82no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4111yescharacteristic
269M16—44no—
270M16—44no—
271M16—44no—
272M16—82no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—44no—
280C2×C8—42no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4C2×C4⋊C414yes—
293C4⋊C4C2×C4⋊C414yes—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—44no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—42no—
301C42—44no—
302C42—44no—
303C42C2×C4⋊C411yescharacteristic
304C42C2×C4⋊C411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42#12011yescharacteristic
310C42#12011yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C23×C411yescharacteristic
316C42C22×D411yescharacteristic
317C42C22×D411yescharacteristic
318C42C22×Q811yescharacteristic
319C23—41no—
320C23—41no—
321C23—41no—
322C23#38511yescharacteristic
323C23#38511yescharacteristic
324C23C2×#12311yescharacteristic
325Q8—44no—
326Q8—44no—
327Q8—44no—
328Q8—82no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
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×C22⋊Q811yescharacteristic
366C8—42no—
367C8—42no—
368C8—44no—
369C8—42no—
370C8—42no—
371C8—44no—
372C8—82no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—82no—
383C22C2×#38511yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—44no—
394C4—44no—
395C4—84no—
396C4#295512yes—
397C4#295612yes—
398C4#302012yes—
399C2—41no—
400C2—41no—
401C2—41no—
402C2#2085711yescharacteristic
403C2#2088411yescharacteristic
404C2#2088511yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24#123
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. #8936C2semidirect
7. #8936C2semidirect
36. #1561C4semidirect
37. #1556C4semidirect
38. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#218442
#178232
#178082
#177392
#89362
#78702
#78682