Finite Group Databasebeta

← #116898

Subgroups — #116898

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21961C211yesmaximal, characteristic, complemented
2#21847C211yesmaximal, characteristic
3#20654C212yesmaximal
4C2×#1292C211yesmaximal, characteristic, complemented
5#7957C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423C2212yes—
9C22×C4⋊C8C2211yescharacteristic
10C2×#421—22no—
11#2285C2212yes—
12#1984C412yescomplemented
13#1954C412yescomplemented
14#1947—24no—
15#1869—22no—
16#1808—22no—
17#1806—24no—
18#1806—24no—
19#1797—22no—
20#1733—22no—
21#1292—22no—
22#1286—24no—
23C22×#41D411yescharacteristic
24C22×M16—22no—
25C23×C8D412yes—
26C4×#41—22no—
27C4×#41—22no—
28C4×#41—22no—
29C4×#41—22no—
30C2×C4×Q8C2×C411yescharacteristic
31C2×C4×Q8Q811yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×C4×D4C2×C411yescharacteristic
35C2×C4×D4Q811yescharacteristic
36C2×#117—22no—
37C2×#117—21no—
38C2×#117D411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41#431—22no—
42#431—22no—
43C2×#107—24no—
44#423—22no—
45#423—22no—
46#423—22no—
47#423D414yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2×C412yes—
54C2×C4⋊C8D412yes—
55#421—44no—
56#421—44no—
57C2×#104—28no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—24no—
60C2×C4×C8C2×C412yes—
61#384—22no—
62#382—22no—
63#380—22no—
64#380—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4D414yes—
67#340—24no—
68#339—28no—
69#336—24no—
70C2×#41—21no—
71C2×#41—21no—
72C2×#41—22no—
73C2×#41—22no—
74C2×#41—22no—
75C2×#41—22no—
76C2×#41D812yes—
77C2×#41Q1612yes—
78C22×Q8C4⋊C411yescharacteristic
79C22×D4—21no—
80C22×D4C4⋊C411yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—42no—
97C22×C8—82no—
98C22×C8C22⋊C414yes—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—41no—
108C4×D4—42no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—42no—
112C4×D4—41no—
113#117—22no—
114#117—22no—
115#117—42no—
116#117—42no—
117#117—42no—
118#117—42no—
119#117—42no—
120#117—42no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C4⋊C411yescharacteristic
127C2×C4⋊C4C4⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—42no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—24no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42C22×C411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×Q811yescharacteristic
143C2×C42#4112yes—
144C2×C42#4112yes—
145C2×C42#4111yescharacteristic
146#107—84no—
147C4⋊C8—24no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153C4⋊C8—42no—
154#104—416no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—84no—
158C4×C8—24no—
159#95—24no—
160#95—42no—
161C24—21no—
162C24#12111yescharacteristic
163#41—24no—
164#41—24no—
165#41—24no—
166#41—24no—
167#41—42no—
168#41—42no—
169#41—42no—
170#41—42no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8#10712yescomplemented
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4#10712yescomplemented
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—81no—
201C22×C4—82no—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C4⋊C411yescharacteristic
204C22×C4#11711yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12012yes—
207C22×C4#12112yes—
208C22×C4#12111yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4#12311yescharacteristic
212C22×C4#12312yes—
213C22×C4#12411yescharacteristic
214C22×C4#12612yes—
215C22×C4#12711yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C2×Q1611yescharacteristic
218C22×C4#13511yescharacteristic
219C22×C4#13511yescharacteristic
220C22×C4AGL(1,ℤ/8)12yes—
221C22×C4#13712yes—
222M16—44no—
223M16—44no—
224M16—82no—
225M16—82no—
226C2×C8—24no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—44no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—84no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42—44no—
258C42—44no—
259C42C4×D412yes—
260C42C4×D412yes—
261C42#12112yes—
262C42#12112yes—
263C42C22⋊Q812yes—
264C42C22⋊Q812yes—
265C42#12312yes—
266C42#12412yes—
267C23—22no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23#38011yescharacteristic
275C23#39211yescharacteristic
276C23#39711yescharacteristic
277C23#46611yescharacteristic
278C23#46711yescharacteristic
279C23#46911yescharacteristic
280C23#47011yescharacteristic
281Q8—24no—
282Q8—42no—
283D4—24no—
284D4—42no—
285D4—42no—
286D4—42no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4#38512yes—
319C2×C4#38611yescharacteristic
320C2×C4#38812yes—
321C2×C4#39011yescharacteristic
322C2×C4#39311yescharacteristic
323C2×C4#39412yes—
324C2×C4#39612yes—
325C2×C4#39911yescharacteristic
326C2×C4C2×#10711yescharacteristic
327C2×C4#42711yescharacteristic
328C2×C4#42812yes—
329C2×C4#44712yes—
330C2×C4#44912yes—
331C2×C4#45112yes—
332C2×C4#45212yes—
333C2×C4#46611yescharacteristic
334C2×C4#46711yescharacteristic
335C2×C4#46812yes—
336C2×C4#48012yes—
337C2×C4#48212yes—
338C2×C4#48412yes—
339C2×C4#48512yes—
340C8—44no—
341C8—82no—
342C8—84no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—42no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22#143911yescharacteristic
359C22#180811yescharacteristic
360C22#180911yescharacteristic
361C22#199911yescharacteristic
362C22#200111yescharacteristic
363C22#207511yescharacteristic
364C22#207611yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—22no—
370C4—42no—
371C4—81no—
372C4—81no—
373C4—164no—
374C4#185212yes—
375C4#185412yes—
376C4#191111yescharacteristic
377C4#191211yescharacteristic
378C4#203312yes—
379C4#203612yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—82no—
387C2#1107211yescharacteristic
388C2#1109611yescharacteristic
389C2#1110911yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21961C2semidirect
4. C2×#1292C2semidirect
5. #7957C2semidirect
12. #1984C4semidirect
13. #1954C4semidirect
174. C2×Q8#107semidirect
184. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
#219612
#218472
#206542
C2×#12922
#79572