Finite Group Databasebeta

← #386850

Subgroups — #386850

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21175C212yesmaximal
2C2×#1945C211yesmaximal, characteristic
3#21039C212yesmaximal
4#11340C212yesmaximal
5#11249C212yesmaximal
6#11242C212yesmaximal
7#10891C212yesmaximal
8#10859C211yesmaximal, characteristic
9#10841C211yesmaximal, characteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#428C2212yes—
13C2×#427C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#424C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2317C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2274C2212yes—
21#1949—22no—
22#1949—22no—
23#1948—22no—
24#1947—22no—
25#1945C418yescomplemented
26#1921—22no—
27#1920—22no—
28#1919—22no—
29#1918—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C2×#117C2312yes—
45C2×#117C2312yes—
46C22×C4⋊C4—41no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C42C2311yescharacteristic
51C2×#108—22no—
52C2×#108—41no—
53C2×#108—41no—
54C2×#108—41no—
55C2×#108—41no—
56C2×#108C2×C414yes—
57#428—42no—
58#428—42no—
59#427—41no—
60#427—41no—
61#427C2×C414yes—
62#427Q814yes—
63C2×#107—22no—
64C2×#107D411yescharacteristic
65C2×#107D411yescharacteristic
66C2×#107D411yescharacteristic
67C2×#107D411yescharacteristic
68C2×#106—22no—
69C2×#106D412yes—
70C2×#106D412yes—
71#424—24no—
72#424C2×C418yes—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4×C8—41no—
76C2×C4×C8—41no—
77C2×C4×C8C2×C414yes—
78#389—22no—
79#389—22no—
80#385—42no—
81#384—22no—
82#384—22no—
83#384—41no—
84#384—41no—
85#384Q814yes—
86#382—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C22⋊C4—42no—
92#343—42no—
93#343—42no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#339—24no—
105#339C2×C418yes—
106#336—22no—
107#336—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16C22×C414yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117C22×C414yes—
149#117C2×Q814yes—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×Q814yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C22×C412yes—
184C2×C42C2×Q812yes—
185C2×C42#4112yes—
186#108—44no—
187#108—44no—
188#107—41no—
189#107—41no—
190#107—44no—
191#107C4⋊C414yescomplemented
192#107C4⋊C414yescomplemented
193#106—42no—
194#106—44no—
195#106C4⋊C418yescomplemented
196C4⋊C8—28no—
197C4⋊C8—28no—
198C4×C8—44no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—42no—
203C24H(5,𝔽2)11yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4C22×Q811yescharacteristic
229C22×C4C22×Q811yescharacteristic
230C22×C4C2×#4111yescharacteristic
231C22×C4C2×#4111yescharacteristic
232C22×C4H(5,𝔽2)11yescharacteristic
233C22×C4H(5,𝔽2)11yescharacteristic
234C22×C4H(5,𝔽2)11yescharacteristic
235C22×C42−1+411yescharacteristic
236C22×C42−1+411yescharacteristic
237C22×C42−1+411yescharacteristic
238C22×C42−1+411yescharacteristic
239M16—24no—
240M16—24no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16—42no—
246M16—42no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8C2×C4⋊C412yes—
270C2×C8C2×C4⋊C412yes—
271C2×C8C4×D412yes—
272C2×C8C4×D412yes—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—88no—
288C4⋊C4—88no—
289C4⋊C4—162no—
290C4⋊C4—162no—
291C4⋊C4C2×C4⋊C414yes—
292C4⋊C4C2×C4⋊C414yes—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C42—24no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—24no—
302C42—42no—
303C42—42no—
304C42—162no—
305C42—162no—
306C42C4×Q814yes—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23#52011yescharacteristic
312C23#53811yescharacteristic
313C23#54311yescharacteristic
314C23D4211yescharacteristic
315C23#55411yescharacteristic
316C23#56211yescharacteristic
317C23#56311yescharacteristic
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—162no—
346C2×C4—162no—
347C2×C4C22×C4⋊C411yescharacteristic
348C2×C4C2×C4×D411yescharacteristic
349C2×C4C2×C4×Q811yescharacteristic
350C2×C4#51811yescharacteristic
351C2×C4#51911yescharacteristic
352C2×C4#52011yescharacteristic
353C2×C4#53411yescharacteristic
354C2×C4#53711yescharacteristic
355C2×C4#53711yescharacteristic
356C2×C4#53711yescharacteristic
357C2×C4#53811yescharacteristic
358C2×C4#53911yescharacteristic
359C2×C4#53911yescharacteristic
360C2×C4#54411yescharacteristic
361C2×C4#54411yescharacteristic
362C2×C4#54411yescharacteristic
363C2×C4#54711yescharacteristic
364C2×C4#54711yescharacteristic
365C2×C4D4×Q811yescharacteristic
366C2×C4D4×Q811yescharacteristic
367C2×C4D4×Q811yescharacteristic
368C2×C4#55011yescharacteristic
369C2×C4#55611yescharacteristic
370C2×C4#55711yescharacteristic
371C2×C4#55711yescharacteristic
372C2×C4Q8211yescharacteristic
373C2×C4#56311yescharacteristic
374C2×C4#56411yescharacteristic
375C8—22no—
376C8—22no—
377C8—22no—
378C8—22no—
379C8—44no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22#232911yescharacteristic
394C22D4×C4⋊C411yescharacteristic
395C22#238211yescharacteristic
396C22#268211yescharacteristic
397C22#268311yescharacteristic
398C22#280511yescharacteristic
399C22#280611yescharacteristic
400C22#329111yescharacteristic
401C22#329111yescharacteristic
402C22#329211yescharacteristic
403C22#329311yescharacteristic
404C4—28no—
405C4—44no—
406C4—44no—
407C4—84no—
408C4—82no—
409C4—82no—
410C4—162no—
411C4#231611yescharacteristic
412C4#232511yescharacteristic
413C4Q8×C4⋊C411yescharacteristic
414C4#236511yescharacteristic
415C4#272611yescharacteristic
416C4#272611yescharacteristic
417C4#272711yescharacteristic
418C4#273411yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2#1444111yescharacteristic
424C2#2268911yescharacteristic
425C2#2269211yescharacteristic
426C2#3122811yescharacteristic
427C2#3123011yescharacteristic
428C2#3125411yescharacteristic
429C2#3125611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1945C4semidirect
191. #107C4⋊C4semidirect
192. #107C4⋊C4semidirect
195. #106C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211752
C2×#19452
#210392
#113402
#112492
#112422
#108912
#108592
#108412