Finite Group Databasebeta

← #193413

Subgroups — #193413

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic
3#17823C211yesmaximal, characteristic
4#17819C211yesmaximal, characteristic
5#17715C211yesmaximal, characteristic
6#8936C218yesmaximal, complemented
7#7869C211yesmaximal, characteristic
8#7868C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2987—22no—
11#2986C2214yes—
12#2985—22no—
13#2985—22no—
14Q8×M16—22no—
15Q8×M16—22no—
16#2921—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1994C2214yes—
24#1986—22no—
25#1826—22no—
26#1574—22no—
27#1574C2214yes—
28#1574C2214yes—
29#1572—22no—
30#1569—22no—
31#1568—22no—
32#1552C414yescomplemented
33#1549C412yescomplemented
34#1546C412yescomplemented
35#1544C2214yes—
36#1276C2214yes—
37#1276C2214yes—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533D412yes—
44#533Q812yes—
45#533C2312yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125Q812yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×#117—21no—
54C2×#117D412yes—
55C22×C42C2311yescharacteristic
56#446—28no—
57#446—28no—
58#446—24no—
59#446—24no—
60C8×Q8—24no—
61C8×Q8—24no—
62#433—28no—
63#433—28no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432C2×C412yes—
70#432D412yes—
71#432D412yes—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431C2×C412yes—
80#431C2312yes—
81#431C2312yes—
82#431C2312yes—
83#431C2312yes—
84#423—24no—
85#423C2×C412yes—
86#423D412yes—
87#423D412yes—
88#423C2312yes—
89C2×C4⋊C8C2×C412yes—
90#407—21no—
91#407—21no—
92#407—21no—
93#407—21no—
94C2×C22⋊C8—21no—
95C4×M16—24no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16—21no—
100C2×C4×C8—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105#376—21no—
106C43C2312yes—
107#339—28no—
108#335—24no—
109#335—24no—
110#334—24no—
111#334—24no—
112#332C2×C418yes—
113#332C2×C418yes—
114C22×Q8—21no—
115C22×Q8—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C22×C8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8C4⋊C412yes—
129C4⋊Q8C4⋊C412yes—
130C4⋊Q8C22×C412yes—
131C4⋊Q8C22×C412yes—
132#125—22no—
133#125—22no—
134#125C4⋊C414yes—
135#125C2×Q814yes—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152#117—22no—
153#117—22no—
154#117—44no—
155#117C4⋊C414yes—
156#117C2×D414yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C4⋊C412yes—
163C2×C4⋊C4C4⋊C412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182C4⋊C8—28no—
183C4⋊C8—28no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8C22×C414yes—
191C4⋊C8C22×C414yes—
192C4⋊C8C2×D414yes—
193C4⋊C8C2×D414yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—28no—
205#97—22no—
206#97—22no—
207#97—22no—
208#97—41no—
209C4×C8—28no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—41no—
216#95—41no—
217#95—41no—
218C24#12311yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—82no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—44no—
250M16—44no—
251M16—44no—
252M16—82no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—82no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—44no—
270C4⋊C4C2×C4⋊C414yes—
271C4⋊C4C2×C4⋊C414yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—42no—
279C42—44no—
280C42—44no—
281C42C2×C4⋊C411yescharacteristic
282C42C2×C4⋊C411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12011yescharacteristic
288C42#12011yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C23×C411yescharacteristic
294C42C22×D411yescharacteristic
295C42C22×D411yescharacteristic
296C42C22×Q811yescharacteristic
297C23—41no—
298C23—41no—
299C23—41no—
300C23#38511yescharacteristic
301C23#38511yescharacteristic
302C23C2×#12311yescharacteristic
303Q8—44no—
304Q8—44no—
305Q8—82no—
306Q8—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38511yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4C22×C4⋊C411yescharacteristic
339C2×C4C2×C4×D411yescharacteristic
340C2×C4C2×C4×D411yescharacteristic
341C2×C4C2×#12011yescharacteristic
342C2×C4C2×C22⋊Q811yescharacteristic
343C2×C4C2×C22⋊Q811yescharacteristic
344C8—44no—
345C8—44no—
346C8—44no—
347C8—44no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—82no—
359C22C2×#38511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—44no—
370C4—44no—
371C4—84no—
372C4#295612yes—
373C4#295712yes—
374C4#302012yes—
375C2—41no—
376C2—41no—
377C2—41no—
378C2#2085711yescharacteristic
379C2#2088511yescharacteristic
380C2#2088611yescharacteristic
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
32. #1552C4semidirect
33. #1549C4semidirect
34. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#602492
#218442
#178232
#178192
#177152
#89362
#78692
#78682