Finite Group Databasebeta

← #220263

Subgroups — #220263

Order 512.

921 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3339C211yesmaximal, characteristic
2#22618C212yesmaximal, complemented
3#22535C212yesmaximal, complemented
4#9785C212yesmaximal, complemented
5#9261C214yesmaximal, complemented
6#8608C211yesmaximal, characteristic, complemented
7#8378C212yesmaximal, complemented
8#8160C211yesmaximal, characteristic, complemented
9C2×#553C2212yes—
10#3339—24no—
11#3339—22no—
12#3339—22no—
13C2×#499C411yescharacteristic, complemented
14C2×#481C2212yes—
15C2×#482C412yescomplemented
16C2×#478C412yescomplemented
17C2×#466C411yescharacteristic, complemented
18C2×#466C2211yescharacteristic
19C2×C4×D8C411yescharacteristic, complemented
20C2×C8×D4C2211yescharacteristic
21C2×#427C411yescharacteristic, complemented
22C22×#107C2211yescharacteristic
23#2345C2214yes—
24#1964—24no—
25#1964C2214yes—
26#1943—22no—
27#1918—24no—
28#1824C2214yes—
29#1807C2212yes—
30#1799—22no—
31#1799C2212yes—
32#1776—22no—
33C4×#99—22no—
34C4×#99C2211yescharacteristic
35#1752C2212yes—
36#1519C2214yes—
37#1459C2211yescharacteristic
38#1289C2212yes—
39#1282C2212yes—
40C22×D8C2×C411yescharacteristic
41C23×C8C2×C411yescharacteristic
42C23×C8C2311yescharacteristic
43#553—24no—
44#553—24no—
45#553—44no—
46C2×#126—22no—
47C2×#125C2×C412yes—
48C2×#123—22no—
49C2×#121—22no—
50C2×#121C2×C412yes—
51C2×#121C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2311yescharacteristic
55C2×#117C2×C412yes—
56C22×C4⋊C4C2312yes—
57#499—22no—
58#499—22no—
59#482—24no—
60#482—24no—
61#481—24no—
62#481—24no—
63#478—24no—
64#478—24no—
65#466—22no—
66#466—22no—
67#466—22no—
68#466—22no—
69C4×D8—22no—
70C4×D8—21no—
71C4×D8—21no—
72C8×D4—22no—
73C8×D4—22no—
74#427—22no—
75#427—21no—
76#427—21no—
77C2×#107C2×C411yescharacteristic
78C2×#107C2×C412yes—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107D414yes—
82C2×#107D412yes—
83C2×#107D412yes—
84C2×#106—22no—
85C2×#106C2×C412yes—
86C2×#106C2×C412yes—
87C2×C4⋊C8C2×C411yescharacteristic
88C2×#102C2×C412yes—
89C2×#102C2×C412yes—
90C2×#102C2×C412yes—
91C2×#99—24no—
92C2×#99—22no—
93C2×#99—24no—
94C2×#99D412yes—
95C2×C22⋊C8C2×C411yescharacteristic
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2311yescharacteristic
98C2×#97—22no—
99C2×C4×C8C2×C411yescharacteristic
100#389—24no—
101#387—44no—
102#380—24no—
103#380C2314yes—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4C2312yes—
107#342—22no—
108#342D412yes—
109#340—22no—
110#340—24no—
111#340—24no—
112#336—22no—
113#327—44no—
114#324—21no—
115#323—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22×C412yes—
119C22×D4C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C22×C411yescharacteristic
123C23×C4C2411yescharacteristic
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—42no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C411yescharacteristic
135C22×C8C22×C411yescharacteristic
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138#126—44no—
139#126—44no—
140#125—24no—
141#125—42no—
142#123—44no—
143#123—44no—
144#121—24no—
145#121—24no—
146#121—24no—
147#121—42no—
148#121—44no—
149#121—44no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—42no—
155C4×D4—44no—
156C4×D4—44no—
157#117—22no—
158#117—22no—
159#117—42no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4#4111yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4C2×D414yes—
178C2×C22⋊C4C2×D411yescharacteristic
179C2×C22⋊C4#4111yescharacteristic
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—44no—
183C2×C42C22×C411yescharacteristic
184#107—21no—
185#107—21no—
186#107C22⋊C412yes—
187#107C22⋊C411yescharacteristic
188#107C22⋊C411yescharacteristic
189#107C22⋊C412yes—
190#107C22⋊C412yes—
191#107C22⋊C414yes—
192#107C22⋊C414yes—
193#106—22no—
194#106—22no—
195#106—24no—
196#106—44no—
197C4⋊C8—22no—
198#102—24no—
199#102—22no—
200#102—22no—
201#102—24no—
202#99—42no—
203#99—48no—
204#99—84no—
205#99—82no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210#97—44no—
211C4×C8—21no—
212C4×C8—21no—
213#95—22no—
214#95—24no—
215#95—44no—
216#95—44no—
217C24—22no—
218C24C4×D411yescharacteristic
219C24C22×D411yescharacteristic
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—162no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—162no—
249C22×C4—162no—
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255C22×C4C2×#4111yescharacteristic
256C22×C4H(5,𝔽2)12yes—
257D8—42no—
258D8—42no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8C2×C22⋊C411yescharacteristic
273C2×C8C2×C22⋊C412yes—
274C2×C8C2×C22⋊C411yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—164no—
287C4⋊C4C2×C22⋊C412yes—
288C4⋊C4C2×C22⋊C412yes—
289C4⋊C4C2×C22⋊C414yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C42—21no—
303C42—21no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—164no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—82no—
320C23—82no—
321C23C2×C4×D411yescharacteristic
322C23D4212yes—
323C23#57512yes—
324C23#57712yes—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—84no—
330D4—84no—
331D4—164no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—162no—
365C2×C4—162no—
366C2×C4C22×C22⋊C411yescharacteristic
367C2×C4#51811yescharacteristic
368C2×C4#54612yes—
369C8—21no—
370C8—22no—
371C8—41no—
372C8—41no—
373C8—82no—
374C8#41012yes—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22D4×C22⋊C411yescharacteristic
392C22#294311yescharacteristic
393C22#295211yescharacteristic
394C22#327114yes—
395C4—22no—
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—84no—
404C4—82no—
405C4—164no—
406C4C2×#41011yescharacteristic
407C4#288411yescharacteristic
408C2—24no—
409C2—21no—
410C2—21no—
411C2—81no—
412C2—162no—
413C2#2240611yescharacteristic
414C2#2244111yescharacteristic
415C2#2245511yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3339C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22618C2semidirect
3. #22535C2semidirect
4. #9785C2semidirect
5. #9261C2semidirect
6. #8608C2semidirect
7. #8378C2semidirect
8. #8160C2semidirect
13. C2×#499C4semidirect
15. C2×#482C4semidirect
16. C2×#478C4semidirect
17. C2×#466C4semidirect
19. C2×C4×D8C4semidirect
21. C2×#427C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33392
#226182
#225352
#97852
#92612
#86082
#83782
#81602