Finite Group Databasebeta

← #120080

Subgroups — #120080

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic
2#14677C211yesmaximal, characteristic
3#10843C212yesmaximal
4C2×#1295C212yesmaximal
5C2×#1295C211yesmaximal, characteristic
6C22×#107—21no—
7C22×#106—21no—
8C22×#97C2212yes—
9C22×#97C2211yescharacteristic
10C2×#384C2211yescharacteristic
11C2×#382C2212yes—
12C2×#382C2211yescharacteristic
13#1847—22no—
14#1847C414yescomplemented
15C2×#340—22no—
16C2×#340—22no—
17#1444—22no—
18#1442—22no—
19#1441—21no—
20#1441—21no—
21#1295—24no—
22#1295—22no—
23C23×C8—21no—
24C23×C8—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×#107—24no—
35C2×#107—42no—
36C2×#107—42no—
37C2×#106—24no—
38C2×#106—42no—
39C2×#106—42no—
40C2×#97—21no—
41C2×#97—21no—
42C2×#97—21no—
43C2×#97—22no—
44C2×#97—22no—
45C2×#97—22no—
46C2×#97—22no—
47C2×#97—22no—
48C2×#97C2×C412yes—
49C2×#97D412yes—
50C2×#97Q814yes—
51C2×#97Q812yes—
52#384—21no—
53#384—22no—
54#384C2×C412yes—
55#382—22no—
56#382—22no—
57#382—22no—
58#382—22no—
59#382—22no—
60#382C2×C412yes—
61#382D412yes—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—22no—
68C2×#95—22no—
69#340—44no—
70#340—44no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—41no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—24no—
86C22×C8—24no—
87C22×C8—24no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4—81no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42C2×Q812yes—
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4112yes—
132C2×C42#4112yes—
133#107—48no—
134#106—48no—
135#97—22no—
136#97—22no—
137#97—22no—
138#97—24no—
139#97—24no—
140#97—42no—
141#97—42no—
142#97C4⋊C414yescomplemented
143#95—24no—
144#95—24no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—81no—
150#95—81no—
151#95—82no—
152#95—82no—
153C24#12011yescharacteristic
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—82no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×D412yes—
193C22×C4#12111yescharacteristic
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q812yes—
198C22×C4#12312yes—
199C22×C4#12311yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4C4⋊Q812yes—
202C22×C4C4⋊Q811yescharacteristic
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—28no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C42C2×C4⋊C411yescharacteristic
236C42#11711yescharacteristic
237C42C4×Q812yes—
238C42#12111yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q812yes—
241C42C22⋊Q812yes—
242C42#12512yes—
243C42#12512yes—
244C42#12612yes—
245C23#38611yescharacteristic
246C23#39411yescharacteristic
247C23#39511yescharacteristic
248C23#45312yes—
249C23#45311yescharacteristic
250C23#45311yescharacteristic
251C23#45412yes—
252C23#45512yes—
253C23#45612yes—
254C23#45611yescharacteristic
255C23#45611yescharacteristic
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—161no—
307C2×C4—162no—
308C2×C4—161no—
309C2×C4#38011yescharacteristic
310C2×C4#38712yes—
311C2×C4#38912yes—
312C2×C4#39111yescharacteristic
313C2×C4#39612yes—
314C2×C4#39811yescharacteristic
315C2×C4#39812yes—
316C2×C4#40011yescharacteristic
317C8—28no—
318C8—48no—
319C22#144111yescharacteristic
320C22#165511yescharacteristic
321C22#165612yes—
322C22#165711yescharacteristic
323C22#165911yescharacteristic
324C22#166012yes—
325C22#166011yescharacteristic
326C22#166112yes—
327C22#166112yes—
328C22#166312yes—
329C22#166412yes—
330C22#188212yes—
331C22#188312yes—
332C22#188611yescharacteristic
333C22#188711yescharacteristic
334C22#203712yes—
335C22#203812yes—
336C22#203911yescharacteristic
337C22#203911yescharacteristic
338C22#206011yescharacteristic
339C22#206012yes—
340C22#206011yescharacteristic
341C22#206112yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—82no—
351C4—82no—
352C4—162no—
353C4—162no—
354C4—162no—
355C2#900012yes—
356C2#900111yescharacteristic
357C2#900311yescharacteristic
358C2#1042411yescharacteristic
359C2#1042712yes—
360C2#1042911yescharacteristic
361C2#1058311yescharacteristic
362C2#1058411yescharacteristic
363C2#1058512yes—
364C2#1121011yescharacteristic
365C2#1121712yes—
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1847C4semidirect
142. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18472
#146772
#108432
C2×#12952