Finite Group Databasebeta

← #100337

Subgroups — #100337

Order 512.

447 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic
4#9694C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#7050C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#477—21no—
9C2×#446—21no—
10#2868C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1987—21no—
13#1947—21no—
14#1947C412yes—
15#1940—22no—
16#1939—21no—
17#1842—21no—
18#1842C2211yescharacteristic
19#1842C2211yescharacteristic
20#1842C2211yescharacteristic
21#1803—21no—
22#1803C412yes—
23#1767—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#332—22no—
27C2×Q8⋊C8—21no—
28#1296—21no—
29#1296—21no—
30#1279—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37#477—42no—
38#477—42no—
39#446—42no—
40#446—42no—
41C2×#107—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yes—
53C2×C4⋊C8D411yescharacteristic
54C2×C4⋊C8D411yescharacteristic
55C2×C4⋊C8D412yes—
56C2×#103—22no—
57C2×#97—22no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384C2×C412yes—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4C2×C411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C43—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—22no—
74#336—22no—
75#336—21no—
76#336—21no—
77#332—44no—
78Q8⋊C8—42no—
79C22×Q8—41no—
80C22×Q8C22⋊C411yescharacteristic
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8—41no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—82no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—22no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—41no—
102C4×Q8SD1612yes—
103C4×Q8Q1612yes—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C22×C411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129#107—44no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—41no—
137C4⋊C8—44no—
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—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—81no—
151C4⋊C8—82no—
152C4⋊C8Q1612yes—
153#103—44no—
154#97—44no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159#95—21no—
160#95—21no—
161#95—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8#10312yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4#10112yes—
221C4⋊C4#10312yes—
222C42—21no—
223C42—21no—
224C42—22no—
225C42—22no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—42no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—81no—
239C42C2×SD1611yescharacteristic
240C42C2×Q1611yescharacteristic
241C42C2×Q1611yescharacteristic
242C42#13511yescharacteristic
243C42AGL(1,ℤ/8)11yescharacteristic
244C42AGL(1,ℤ/8)11yescharacteristic
245C42AGL(1,ℤ/8)11yescharacteristic
246C42AGL(1,ℤ/8)11yescharacteristic
247C42#13711yescharacteristic
248C42#13711yescharacteristic
249C42#13711yescharacteristic
250C42#13711yescharacteristic
251C23#38611yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—41no—
255Q8—41no—
256Q8—162no—
257Q8—162no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4C2×#10111yescharacteristic
283C2×C4#41311yescharacteristic
284C2×C4C2×#10311yescharacteristic
285C2×C4#41711yescharacteristic
286C2×C4#41711yescharacteristic
287C2×C4#41811yescharacteristic
288C2×C4C4×Q1611yescharacteristic
289C2×C4#44111yescharacteristic
290C2×C4#45011yescharacteristic
291C2×C4#45111yescharacteristic
292C2×C4#45111yescharacteristic
293C2×C4#45211yescharacteristic
294C2×C4#45311yescharacteristic
295C2×C4#45411yescharacteristic
296C2×C4#46211yescharacteristic
297C2×C4#46411yescharacteristic
298C2×C4#46811yescharacteristic
299C2×C4#46811yescharacteristic
300C2×C4#46911yescharacteristic
301C2×C4#46911yescharacteristic
302C2×C4#47111yescharacteristic
303C2×C4#47311yescharacteristic
304C2×C4#48311yescharacteristic
305C2×C4#48411yescharacteristic
306C8—42no—
307C8—82no—
308C8—82no—
309C8—81no—
310C8—82no—
311C8—161no—
312C22#187211yescharacteristic
313C22#187811yescharacteristic
314C22#188311yescharacteristic
315C22#189511yescharacteristic
316C22#190111yescharacteristic
317C22#190311yescharacteristic
318C22#190811yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—81no—
330C4—162no—
331C4#154911yescharacteristic
332C4#155011yescharacteristic
333C4#155711yescharacteristic
334C4#156111yescharacteristic
335C4#163111yescharacteristic
336C4#163311yescharacteristic
337C4#163611yescharacteristic
338C4#163711yescharacteristic
339C4#169211yescharacteristic
340C4#169411yescharacteristic
341C4#170011yescharacteristic
342C4#170211yescharacteristic
343C2#897511yescharacteristic
344C2#913711yescharacteristic
345C2#916911yescharacteristic
346C2#922611yescharacteristic
347C2#925011yescharacteristic
348C2#932211yescharacteristic
349C2#935711yescharacteristic
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
#331982
#227222
#98962
#96942
#85642
#70502
#70312