Finite Group Databasebeta

← #217069

Subgroups — #217069

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9904C211yesmaximal, characteristic, complemented
5#9899C211yesmaximal, characteristic, complemented
6#9706C211yesmaximal, characteristic, complemented
7#9695C211yesmaximal, characteristic, complemented
8#9134C212yesmaximal, complemented
9#9133C212yesmaximal, complemented
10#8588C212yesmaximal, complemented
11#8585C212yesmaximal, complemented
12C2×#540—21no—
13C2×#433—21no—
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2867C2211yescharacteristic
17#2859C2211yescharacteristic
18#2856C2212yes—
19#2641—21no—
20#2619—21no—
21#2445—22no—
22#2445C414yes—
23#2428—22no—
24#2428C414yes—
25#2328—21no—
26C4×#127C2212yes—
27C4×#117C2211yescharacteristic
28#1939—22no—
29#1850—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1832—21no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1798—21no—
37#1797C2212yes—
38#1768—21no—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C2×#331C2212yes—
42C2×#331C2212yes—
43C2×#331C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C23×C8—21no—
50#540—44no—
51#540—44no—
52C2×#127—21no—
53C2×#127—21no—
54C2×#127—22no—
55C2×#127C2×C414yes—
56C2×#127C2312yes—
57C2×#125—21no—
58C2×#123—24no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—24no—
62C2×#121C2×C414yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42C2311yescharacteristic
69#433—44no—
70#431—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×#97—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#394—24no—
89#394—24no—
90#394—24no—
91#392—24no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390C2×C414yes—
97#390C2×C414yes—
98#389—22no—
99#389C2×C414yes—
100#386—24no—
101#386—24no—
102#384C2312yes—
103#382—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4D412yes—
108#376D411yescharacteristic
109#376D411yescharacteristic
110C43C2311yescharacteristic
111C43C2311yescharacteristic
112#336—22no—
113#336—21no—
114#336—21no—
115#336—22no—
116#336—22no—
117#331—42no—
118#331—42no—
119#331—42no—
120#331—42no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22×C414yes—
127C23×C4—24no—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—42no—
138C22×C8—82no—
139#127—22no—
140#127—22no—
141#127—22no—
142#127—22no—
143#127—44no—
144#127—42no—
145#127—42no—
146#127—82no—
147#127—82no—
148#127—84no—
149#125—42no—
150#125—42no—
151#123—416no—
152#121—44no—
153#121—44no—
154#121—416no—
155#121—82no—
156#121—82no—
157#121—84no—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—82no—
161C4×D4—82no—
162C4×D4—84no—
163#117—42no—
164#117D814yes—
165#117SD1614yes—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C414yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C412yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—44no—
203C4⋊C8—42no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8—82no—
207C22⋊C8—42no—
208C22⋊C8—44no—
209#97—42no—
210C4×C8—42no—
211C4×C8—42no—
212#95—24no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217C24—24no—
218C24C2×C22⋊C411yescharacteristic
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—48no—
224C2×D4—48no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—48no—
238C2×D4—84no—
239C2×D4—84no—
240C2×D4—84no—
241C2×D4—84no—
242C2×D4—84no—
243C2×D4—84no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×D812yes—
264C22×C4C2×SD1612yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—84no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4#10218yes—
291C22⋊C4—42no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C22⋊C4#10218yes—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C42C2×D812yes—
312C42C2×D812yes—
313C42C2×SD1612yes—
314C42C2×SD1612yes—
315C42H(5,𝔽2)12yes—
316C422−1+412yes—
317C23—21no—
318C23—22no—
319C23—21no—
320C23—22no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23C2×#10212yes—
329C23C22×C22⋊C411yescharacteristic
330D4—84no—
331D4—84no—
332D4—82no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—82no—
340D4—168no—
341D4—168no—
342D4—168no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—82no—
369C2×C4—168no—
370C2×C4C2×#10214yes—
371C2×C4C2×#10214yes—
372C2×C4C2×#10212yes—
373C2×C4#52012yes—
374C2×C4#53511yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4#53712yes—
377C2×C4C22×D811yescharacteristic
378C2×C4C22×SD1611yescharacteristic
379C2×C4#57512yes—
380C2×C4#57612yes—
381C2×C4#57712yes—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8—82no—
387C8—82no—
388C22—24no—
389C22—22no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—84no—
397C22—84no—
398C22—84no—
399C22—84no—
400C22#231111yescharacteristic
401C22#288812yes—
402C22C22×#10211yescharacteristic
403C22#289611yescharacteristic
404C22#289712yes—
405C4—24no—
406C4—24no—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—84no—
414C4#322011yescharacteristic
415C4#322111yescharacteristic
416C4#324112yes—
417C4#324512yes—
418C4#324612yes—
419C2—42no—
420C2—164no—
421C2#1772012yes—
422C2#1776612yes—
423C2#2187211yescharacteristic
424C2#2188211yescharacteristic
425C2#2188411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35278C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21972C2semidirect
3. #20812C2semidirect
4. #9904C2semidirect
5. #9899C2semidirect
6. #9706C2semidirect
7. #9695C2semidirect
8. #9134C2semidirect
9. #9133C2semidirect
10. #8588C2semidirect
11. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#219722
#208122
#99042
#98992
#97062
#96952
#91342
#91332
#85882
#85852