Finite Group Databasebeta

← #242757

Subgroups — #242757

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21970C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4#17808C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6C2×#1546C211yesmaximal, characteristic
7#10608C214yesmaximal, complemented
8#10604C214yesmaximal, complemented
9#8000C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1966—22no—
18#1966C2212yes—
19#1966C2212yes—
20#1950—22no—
21#1950C2214yes—
22#1811—22no—
23#1784—22no—
24#1574—22no—
25#1574—22no—
26#1572—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1569—22no—
30#1565C2214yes—
31#1556—22no—
32#1549—22no—
33#1546—22no—
34#1546—22no—
35#1546C2214yes—
36C2×#332C2214yes—
37#1292C2212yes—
38#1292C2212yes—
39C23×Q8—21no—
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⋊C4Q812yes—
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—
89C2×#101—24no—
90C2×#101—24no—
91#407—21no—
92#407—21no—
93#407—21no—
94#407D412yes—
95C2×C22⋊C8—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16C2312yes—
100C2×C4×C8—21no—
101#341—24no—
102#335—24no—
103#335—24no—
104#334—24no—
105#334—24no—
106#332—28no—
107#332—28no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—82no—
115C22×Q8—82no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q811yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—82no—
127C2×M16C2×D414yes—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C2×Q814yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42C2411yescharacteristic
170#108—84no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8C2×D414yes—
186C4⋊C8C2×D414yes—
187#101—84no—
188#101—84no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—22no—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—41no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C24C22⋊Q811yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—84no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—84no—
222C2×Q8—84no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—84no—
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×Q811yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—24no—
250M16—24no—
251M16—42no—
252M16—42no—
253M16—42no—
254M16—42no—
255M16—82no—
256M16—82no—
257M16—82no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—82no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42#12611yescharacteristic
286C42#12611yescharacteristic
287C42C22×D411yescharacteristic
288C42C22×D411yescharacteristic
289C42C2×#4111yescharacteristic
290C42C2×#4111yescharacteristic
291C23—22no—
292C23—41no—
293C23—41no—
294C23#39611yescharacteristic
295C23#39611yescharacteristic
296C23C2×C22⋊Q811yescharacteristic
297Q8—84no—
298Q8—84no—
299Q8—82no—
300Q8—82no—
301Q8—84no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4#39611yescharacteristic
326C2×C4#39611yescharacteristic
327C2×C4#39611yescharacteristic
328C2×C4#39611yescharacteristic
329C2×C4#39611yescharacteristic
330C2×C4#39611yescharacteristic
331C2×C4C2×#12111yescharacteristic
332C2×C4C2×#12111yescharacteristic
333C2×C4C2×#12111yescharacteristic
334C2×C4C2×C22⋊Q811yescharacteristic
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×#12611yescharacteristic
337C2×C4#57612yes—
338C2×C4#57712yes—
339C8—44no—
340C8—44no—
341C8—44no—
342C8—81no—
343C8—81no—
344C8—82no—
345C8—82no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22C2×#39611yescharacteristic
356C22#308112yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—84no—
366C4#207012yes—
367C4#304111yescharacteristic
368C4#304311yescharacteristic
369C4#306212yes—
370C4#306912yes—
371C2—22no—
372C2—41no—
373C2—41no—
374C2#2360311yescharacteristic
375C2#2361411yescharacteristic
376C2C2×#207011yescharacteristic
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. #10608C2semidirect
8. #10604C2semidirect

Maximal subgroups

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

subgroupindex
#219702
#217992
#178202
#178082
#177422
C2×#15462
#106082
#106042
#80002