Finite Group Databasebeta

← #100309

Subgroups — #100309

Order 512.

471 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2Q8×C4⋊C8C211yesmaximal, characteristic
3#9694C211yesmaximal, characteristic
4#9692C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#7050C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#475—21no—
9C2×C8×Q8—21no—
10#2868C2211yescharacteristic
11C42×Q8C2211yescharacteristic
12#1988—21no—
13#1948—21no—
14#1948C412yes—
15#1939—22no—
16#1939—21no—
17#1842—21no—
18#1842C2211yescharacteristic
19#1842C2211yescharacteristic
20#1842C2211yescharacteristic
21#1803—21no—
22#1803C412yes—
23C4×C4⋊C8—21no—
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#332—22no—
27C2×#332—21no—
28C2×#332—21no—
29C2×Q8⋊C8—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#475—42no—
38#475—42no—
39C8×Q8—42no—
40C8×Q8—42no—
41C2×#106—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×C4×C8—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—
78#332—42no—
79#332—42no—
80Q8⋊C8—42no—
81C22×Q8—41no—
82C22×Q8C22⋊C411yescharacteristic
83C22×C8—21no—
84C22×C8—22no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—22no—
104C4×Q8—41no—
105C4×Q8D812yes—
106C4×Q8SD1612yes—
107C4×Q8SD1612yes—
108C4×Q8Q1612yes—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134#106—44no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141C4⋊C8—41no—
142C4⋊C8—44no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—81no—
156C4⋊C8—82no—
157C4⋊C8SD1612yes—
158#103—44no—
159C4×C8—44no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164#95—21no—
165#95—21no—
166#95—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8#10112yes—
177C2×Q8#10212yes—
178C2×Q8#10312yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12311yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4#10212yes—
229C4⋊C4#10312yes—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—21no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—81no—
247C42C2×D811yescharacteristic
248C42C2×SD1611yescharacteristic
249C42C2×SD1611yescharacteristic
250C42C2×SD1611yescharacteristic
251C42C2×Q1611yescharacteristic
252C42#13511yescharacteristic
253C42#13511yescharacteristic
254C42#13511yescharacteristic
255C42AGL(1,ℤ/8)11yescharacteristic
256C42AGL(1,ℤ/8)11yescharacteristic
257C42#13711yescharacteristic
258C42#13711yescharacteristic
259C23#38611yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—161no—
264Q8—161no—
265Q8—161no—
266Q8—161no—
267Q8#33214yescomplemented
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4C2×#10111yescharacteristic
293C2×C4#41311yescharacteristic
294C2×C4C2×#10211yescharacteristic
295C2×C4C2×#10311yescharacteristic
296C2×C4#41611yescharacteristic
297C2×C4#41711yescharacteristic
298C2×C4C4×SD1611yescharacteristic
299C2×C4#44011yescharacteristic
300C2×C4#44711yescharacteristic
301C2×C4#44811yescharacteristic
302C2×C4#44811yescharacteristic
303C2×C4#44911yescharacteristic
304C2×C4#45311yescharacteristic
305C2×C4#45411yescharacteristic
306C2×C4#46011yescharacteristic
307C2×C4#46411yescharacteristic
308C2×C4#46511yescharacteristic
309C2×C4#46511yescharacteristic
310C2×C4#46711yescharacteristic
311C2×C4#46711yescharacteristic
312C2×C4#47111yescharacteristic
313C2×C4#47311yescharacteristic
314C2×C4#48011yescharacteristic
315C2×C4#48111yescharacteristic
316C8—42no—
317C8—82no—
318C8—82no—
319C8—81no—
320C8—82no—
321C8—161no—
322C22#187411yescharacteristic
323C22#188011yescharacteristic
324C22#188311yescharacteristic
325C22#189211yescharacteristic
326C22#190111yescharacteristic
327C22#190211yescharacteristic
328C22#190811yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—24no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—41no—
338C4—42no—
339C4—81no—
340C4—162no—
341C4C2×#33211yescharacteristic
342C4#154011yescharacteristic
343C4#155011yescharacteristic
344C4#155211yescharacteristic
345C4#162111yescharacteristic
346C4#162311yescharacteristic
347C4#162611yescharacteristic
348C4#162711yescharacteristic
349C4#166611yescharacteristic
350C4#167011yescharacteristic
351C4#168011yescharacteristic
352C4#168411yescharacteristic
353C2#897911yescharacteristic
354C2#913111yescharacteristic
355C2#917111yescharacteristic
356C2#922011yescharacteristic
357C2#925211yescharacteristic
358C2#931711yescharacteristic
359C2#935811yescharacteristic
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

Complements

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

NH ≅ G/Ntype
267. Q8#332semidirect

Maximal subgroups

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

subgroupindex
#331962
Q8×C4⋊C82
#96942
#96922
#85642
#70502
#70312