Finite Group Databasebeta

← #6392794

Subgroups — #6392794

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2C2×#2987C211yesmaximal, characteristic
3#60192C211yesmaximal, characteristic
4#22747C218yesmaximal, complemented
5#22733C218yesmaximal, complemented
6#21995C214yesmaximal
7#21994C212yesmaximal
8#21372C214yesmaximal
9#21371C212yesmaximal
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2987C418yes—
13#2987C412yes—
14#2987C412yes—
15#2987C414yes—
16#2986—22no—
17#2985—24no—
18Q8×M16C2218yes—
19Q8×M16C2214yes—
20Q8×M16C2214yes—
21#2962—24no—
22#2961—22no—
23#2960—22no—
24C2×#446C2218yes—
25#2921C2214yes—
26#2920C2212yes—
27C2×#433C2214yes—
28C2×#432C2212yes—
29#2905C2212yes—
30C2×#423C2211yescharacteristic
31#2873C2212yes—
32C2×C4×M16C2211yescharacteristic
33#1950C2214yes—
34#1950C2214yes—
35#1950C2218yes—
36#1950C2218yes—
37#1950C2218yes—
38#1950C2218yes—
39#1950C2218yes—
40#1855C2214yes—
41#1855C2214yes—
42#1855C2218yes—
43#1849C2214yes—
44#1849C2214yes—
45#1848C2214yes—
46#1838C2214yes—
47#1838C2214yes—
48#1837C2214yes—
49#1778C2218yes—
50#1778C2218yes—
51C4×#108C2218yes—
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2312yes—
55#533C2×C414yes—
56#533C2×C411yescharacteristic
57#533C2×C411yescharacteristic
58#533C2×C412yes—
59#533D412yes—
60#533D412yes—
61#533D414yes—
62C2×C4⋊Q8Q812yes—
63C2×#125Q812yes—
64C2×C22⋊Q8Q814yes—
65C2×C4×Q8C2314yes—
66C2×#117C2312yes—
67C22×C42C2311yescharacteristic
68#446—28no—
69#446—28no—
70#446—28no—
71#446—24no—
72#446—24no—
73#446—28no—
74#446—28no—
75#446C2×C4116yes—
76#446C2×C4116yes—
77#446C2×C4116yes—
78#446C2×C4116yes—
79C8×Q8—28no—
80C8×Q8—24no—
81C8×Q8—24no—
82#433—28no—
83#433—28no—
84#433C2×C418yes—
85#433C2×C418yes—
86#433C2×C418yes—
87#433C2×C418yes—
88#432—22no—
89#432—22no—
90#432C2×C414yes—
91#432C2×C414yes—
92#432C2×C414yes—
93#432C2×C414yes—
94#431—22no—
95#431—22no—
96C2×#108C2318yes—
97C2×#108C2318yes—
98C2×#108C2318yes—
99C2×#108C2318yes—
100#424—24no—
101#424—24no—
102#423—21no—
103#423—21no—
104#423C2314yes—
105#423C2314yes—
106#423C2312yes—
107#423C2314yes—
108#423C2314yes—
109#423C2314yes—
110#423C2312yes—
111C2×C4⋊C8—22no—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2318yes—
114#407C2314yes—
115C2×C22⋊C8C2314yes—
116#405—24no—
117#405—24no—
118C4×M16—22no—
119C4×M16C2314yes—
120C4×M16C2318yes—
121C4×M16C2314yes—
122C4×M16C2318yes—
123C2×#97—21no—
124C2×#97C2314yes—
125C2×#97C2314yes—
126C2×C4×C8—21no—
127C2×C4×C8C2314yes—
128#341C2318yes—
129#341C2318yes—
130#341C2318yes—
131#341C2318yes—
132C22×Q8C2×Q814yes—
133C23×C4C2×Q812yes—
134C23×C4C2411yescharacteristic
135C2×M16—21no—
136C2×M16—24no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—42no—
140C2×M16—44no—
141C2×M16C2412yes—
142C2×M16C2414yes—
143C2×M16C2414yes—
144C2×M16C2414yes—
145C2×M16C2412yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8C2418yes—
150C4⋊Q8C4⋊C414yes—
151C4⋊Q8C4⋊C412yes—
152C4⋊Q8C4⋊C412yes—
153C4⋊Q8C4⋊C414yes—
154C4⋊Q8C4⋊C414yes—
155#125C4⋊C412yes—
156#125C4⋊C412yes—
157#125C4⋊C414yes—
158#125C4⋊C418yes—
159C22⋊Q8C4⋊C418yes—
160C22⋊Q8C4⋊C414yes—
161C22⋊Q8C4⋊C414yes—
162C22⋊Q8C4⋊C414yes—
163C22⋊Q8C4⋊C414yes—
164C22⋊Q8C4⋊C418yes—
165C4×Q8C22×C414yes—
166C4×Q8C22×C414yes—
167C4×Q8C22×C418yes—
168C4×Q8C2×D418yes—
169C4×Q8C2×D414yes—
170C4×Q8C2×D414yes—
171#117C22×C414yes—
172#117C22×C412yes—
173#117C22×C412yes—
174#117C2×D414yes—
175#117C2×D414yes—
176C2×C4⋊C4C2×Q814yes—
177C2×C4⋊C4C2×Q818yes—
178C2×C4⋊C4C2×Q814yes—
179C2×C22⋊C4C2×Q814yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×Q811yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42C2414yes—
189C2×C42C2412yes—
190#108—216no—
191#108—48no—
192#108—48no—
193#108—48no—
194C4⋊C8—24no—
195C4⋊C8—24no—
196C4⋊C8—28no—
197C4⋊C8—28no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—28no—
203C4⋊C8—28no—
204C4⋊C8C22×C418yes—
205C4⋊C8C22×C418yes—
206C4⋊C8C22×C4116yes—
207C4⋊C8C22×C4116yes—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8C22×C418yes—
211C22⋊C8C22×C418yes—
212#97—24no—
213#97—28no—
214#97—24no—
215#97—28no—
216#97C22×C418yes—
217#97C22×C418yes—
218#97C22×C418yes—
219#97C22×C418yes—
220C4×C8—24no—
221C4×C8—28no—
222C4×C8—24no—
223C4×C8—28no—
224C4×C8C22×C418yes—
225C4×C8C22×C418yes—
226C24C22×Q811yescharacteristic
227C2×Q8—84no—
228C2×Q8C2×C4⋊C418yes—
229C2×Q8C2×C4⋊C414yes—
230C2×Q8C2×C4⋊C414yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—82no—
234C22×C4C2×C4⋊C412yes—
235C22×C4C2×C4⋊C412yes—
236C22×C4C2×C4⋊C412yes—
237C22×C4C2×C4⋊C412yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243C22×C4C22×Q812yes—
244C22×C4C22×Q812yes—
245C22×C4C22×Q818yes—
246C22×C4C2511yescharacteristic
247M16—24no—
248M16—22no—
249M16—28no—
250M16—28no—
251M16—24no—
252M16—28no—
253M16—42no—
254M16—44no—
255M162−1+414yes—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—28no—
259C2×C8—28no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—44no—
265C2×C8—42no—
266C2×C8C23×C4116yes—
267C4⋊C4C2×C4⋊C414yes—
268C4⋊C4C2×C4⋊C414yes—
269C4⋊C4C2×C4⋊C418yes—
270C4⋊C4C2×C4⋊C418yes—
271C4⋊C4C2×C4⋊C418yes—
272C4⋊C4C2×C4⋊C418yes—
273C4⋊C4C2×C4⋊C418yes—
274C4⋊C4C2×C4⋊C418yes—
275C4⋊C4C2×C4⋊C414yes—
276C4⋊C4C2×C4⋊C414yes—
277C22⋊C4C2×C4⋊C418yes—
278C22⋊C4C2×C4⋊C414yes—
279C22⋊C4C2×C4⋊C414yes—
280C42C2×C4⋊C411yescharacteristic
281C42C2×C4⋊C411yescharacteristic
282C42C2×C4⋊C412yes—
283C42C2×C4⋊C411yescharacteristic
284C42C2×C4⋊C411yescharacteristic
285C42C2×C4⋊C412yes—
286C42C23×C414yes—
287C42C23×C414yes—
288C42C23×C412yes—
289C42C23×C412yes—
290C42C22×D412yes—
291C42C22×D412yes—
292C42C22×D418yes—
293C23—22no—
294C23—42no—
295C23C22×C4⋊C411yescharacteristic
296C23C22×C4⋊C411yescharacteristic
297C23C23×Q811yescharacteristic
298Q8—48no—
299Q8—44no—
300Q8—44no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—48no—
315C2×C4C22×C4⋊C411yescharacteristic
316C2×C4C22×C4⋊C411yescharacteristic
317C2×C4C22×C4⋊C412yes—
318C2×C4C22×C4⋊C412yes—
319C2×C4C22×C4⋊C412yes—
320C2×C4C22×C4⋊C412yes—
321C2×C4C22×C4⋊C418yes—
322C2×C4C22×C4⋊C418yes—
323C2×C4C24×C411yescharacteristic
324C2×C4C23×D411yescharacteristic
325C2×C4C2×2−1+412yes—
326C2×C4#58512yes—
327C8—216no—
328C8—24no—
329C8—44no—
330C8—42no—
331C8—44no—
332C8—44no—
333C22—22no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—82no—
339C22C23×C4⋊C411yescharacteristic
340C22#350212yes—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—28no—
348C4—28no—
349C4C4×2−1+412yes—
350C4#348611yescharacteristic
351C4#348711yescharacteristic
352C2—22no—
353C2—41no—
354C2—41no—
355C2#3424911yescharacteristic
356C2#6011312yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C4⋊C4
prime radicalC24C22×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22747C2semidirect
5. #22733C2semidirect

Maximal subgroups

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

subgroupindex
#602492
C2×#29872
#601922
#227472
#227332
#219952
#219942
#213722
#213712