Finite Group Databasebeta

← #242759

Subgroups — #242759

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21970C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4#17808C211yesmaximal, characteristic
5#17746C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10609C214yesmaximal, complemented
8#10605C214yesmaximal, complemented
9#8000C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1997—22no—
18#1997C2212yes—
19#1997C2212yes—
20#1950—22no—
21#1950C2214yes—
22#1821—22no—
23#1811—22no—
24#1574—22no—
25#1574—22no—
26#1572—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1569—22no—
30#1565C2214yes—
31#1558—22no—
32#1548—22no—
33#1548—22no—
34#1548—22no—
35#1548C2214yes—
36C2×#333C2214yes—
37#1292C2212yes—
38#1292C2212yes—
39C23×Q8Q812yes—
40C22×M16—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8Q812yes—
52C2×C4⋊Q8C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431C2312yes—
76C2×#108—24no—
77#423—21no—
78#423—21no—
79#423D412yes—
80#423D412yes—
81#423D412yes—
82#423D412yes—
83#423C2312yes—
84#423C2311yescharacteristic
85#423C2311yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89#407—21no—
90#407—21no—
91#407—21no—
92#407D412yes—
93C2×C22⋊C8—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2312yes—
98C2×C4×C8—21no—
99#343—24no—
100#343—24no—
101#341—24no—
102#335—24no—
103#335—24no—
104#334—24no—
105#334—24no—
106#333—28no—
107#333—28no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—84no—
114C22×Q8C2×Q814yes—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×Q811yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—82no—
126C2×M16C2×D414yes—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×Q811yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#108—84no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8C2×D414yes—
185C4⋊C8C2×D414yes—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97—41no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8—41no—
204C4×C8—41no—
205C4×C8—41no—
206C24C22⋊Q811yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×Q8—84no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×Q811yescharacteristic
242C22×C4C2×#4111yescharacteristic
243M16—24no—
244M16—24no—
245M16—42no—
246M16—42no—
247M16—42no—
248M16—42no—
249M16—82no—
250M16—82no—
251M16—82no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—82no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42#12611yescharacteristic
280C42#12611yescharacteristic
281C42C22×D411yescharacteristic
282C42C22×D411yescharacteristic
283C42C2×#4111yescharacteristic
284C42C2×#4111yescharacteristic
285C23—22no—
286C23—41no—
287C23—41no—
288C23#39611yescharacteristic
289C23#39611yescharacteristic
290C23C2×C22⋊Q811yescharacteristic
291Q8—84no—
292Q8—84no—
293Q8—84no—
294Q8—84no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4#39611yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39611yescharacteristic
322C2×C4#39611yescharacteristic
323C2×C4#39611yescharacteristic
324C2×C4C2×#12111yescharacteristic
325C2×C4C2×#12111yescharacteristic
326C2×C4C2×#12111yescharacteristic
327C2×C4C2×C22⋊Q811yescharacteristic
328C2×C4C2×C22⋊Q811yescharacteristic
329C2×C4C2×#12611yescharacteristic
330C2×C4#57712yes—
331C2×C4#57812yes—
332C8—44no—
333C8—44no—
334C8—44no—
335C8—81no—
336C8—81no—
337C8—82no—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22C2×#39611yescharacteristic
349C22#308212yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—84no—
359C4#206912yes—
360C4#304211yescharacteristic
361C4#304411yescharacteristic
362C4#306312yes—
363C4#306912yes—
364C2—22no—
365C2—41no—
366C2—41no—
367C2#2360311yescharacteristic
368C2#2361511yescharacteristic
369C2C2×#206911yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #10609C2semidirect
8. #10605C2semidirect

Maximal subgroups

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

subgroupindex
#219702
#218292
#178202
#178082
#177462
C2×#15482
#106092
#106052
#80002