Finite Group Databasebeta

← #100316

Subgroups — #100316

Order 512.

435 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic
4#9691C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#7050C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#475C412yes—
9#2868C2211yescharacteristic
10C42×Q8C2211yescharacteristic
11#1988C412yes—
12#1984—21no—
13#1948—21no—
14#1948—21no—
15#1916—22no—
16#1842C2211yescharacteristic
17#1842C2211yescharacteristic
18#1842C2211yescharacteristic
19#1831—21no—
20#1803—21no—
21#1803—21no—
22C8×C4⋊C4—21no—
23C4×C4⋊C8—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#333—21no—
27C2×#333—21no—
28C2×Q8⋊C8—21no—
29#1296—22no—
30#1279—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8C2×C412yes—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37#475—44no—
38C2×#106—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C412yes—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D412yes—
51C2×#103—22no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—22no—
55#384—21no—
56#384—21no—
57#384—21no—
58#384—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C43—21no—
66C43C2311yescharacteristic
67#340—22no—
68#340—22no—
69#336—22no—
70#336—22no—
71#336—22no—
72#336—21no—
73#336—21no—
74#333—42no—
75#333—42no—
76Q8⋊C8—42no—
77C22×Q8—22no—
78C22×Q8C22⋊C411yescharacteristic
79C22×C8—21no—
80C22×C8—22no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—82no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—22no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—41no—
100C4×Q8SD1612yes—
101C4×Q8Q1612yes—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42C22×C411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—81no—
148C4⋊C8SD1612yes—
149#103—44no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8—82no—
155#95—21no—
156#95—21no—
157#95—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—82no—
166C2×Q8—82no—
167C2×Q8#10312yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12311yescharacteristic
180C2×C8—22no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4#10012yes—
217C4⋊C4#10312yes—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—22no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—41no—
227C42—42no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—81no—
235C42C2×SD1611yescharacteristic
236C42C2×SD1611yescharacteristic
237C42C2×Q1611yescharacteristic
238C42#13511yescharacteristic
239C42#13511yescharacteristic
240C42#13511yescharacteristic
241C42#13511yescharacteristic
242C42#13511yescharacteristic
243C42AGL(1,ℤ/8)11yescharacteristic
244C42#13711yescharacteristic
245C42#13711yescharacteristic
246C42#13711yescharacteristic
247C23#38611yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—41no—
251Q8—41no—
252Q8—162no—
253Q8—162no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4C2×#10011yescharacteristic
279C2×C4#41311yescharacteristic
280C2×C4C2×#10311yescharacteristic
281C2×C4#41611yescharacteristic
282C2×C4#41611yescharacteristic
283C2×C4#41911yescharacteristic
284C2×C4C4×SD1611yescharacteristic
285C2×C4#44011yescharacteristic
286C2×C4#44811yescharacteristic
287C2×C4#44911yescharacteristic
288C2×C4#45211yescharacteristic
289C2×C4#45211yescharacteristic
290C2×C4#45511yescharacteristic
291C2×C4#45611yescharacteristic
292C2×C4#46011yescharacteristic
293C2×C4#46411yescharacteristic
294C2×C4#46511yescharacteristic
295C2×C4#46511yescharacteristic
296C2×C4#46711yescharacteristic
297C2×C4#46711yescharacteristic
298C2×C4#47111yescharacteristic
299C2×C4#47311yescharacteristic
300C2×C4#48511yescharacteristic
301C2×C4#48511yescharacteristic
302C8—42no—
303C8—82no—
304C8—82no—
305C8—82no—
306C8—81no—
307C8—161no—
308C22#187311yescharacteristic
309C22#188011yescharacteristic
310C22#188211yescharacteristic
311C22#189211yescharacteristic
312C22#190011yescharacteristic
313C22#190311yescharacteristic
314C22#190811yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—42no—
324C4—42no—
325C4—81no—
326C4—162no—
327C4#154211yescharacteristic
328C4#154311yescharacteristic
329C4#154811yescharacteristic
330C4#155411yescharacteristic
331C4#164011yescharacteristic
332C4#164311yescharacteristic
333C4#165011yescharacteristic
334C4#165311yescharacteristic
335C4#166611yescharacteristic
336C4#167011yescharacteristic
337C4#168011yescharacteristic
338C4#168411yescharacteristic
339C2#898211yescharacteristic
340C2#912811yescharacteristic
341C2#917011yescharacteristic
342C2#922011yescharacteristic
343C2#925411yescharacteristic
344C2#931511yescharacteristic
345C2#935311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#331962
#228212
#98962
#96912
#85642
#70502
#70312