Finite Group Databasebeta

← #378918

Subgroups — #378918

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#1943C211yesmaximal, characteristic
3C2×#1771C211yesmaximal, characteristic
4#14045C212yesmaximal
5#11377C211yesmaximal, characteristic
6#11327C211yesmaximal, characteristic
7#11231C211yesmaximal, characteristic
8#10884C212yesmaximal
9#10881C212yesmaximal
10#10880C212yesmaximal
11#10703C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#2086C418yescomplemented
24#1943—24no—
25#1771—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1444—22no—
37#1441—22no—
38#1436—22no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42—22no—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#107—22no—
60C2×#107C2×C414yes—
61C2×#106—22no—
62C2×#106—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2×C414yes—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97—22no—
69C2×#97—42no—
70C2×#97D414yes—
71C2×C4×C8—22no—
72C2×C4×C8—42no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—42no—
78#389C2×C418yes—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—42no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—22no—
95#340—24no—
96#340—42no—
97#340C2×C418yes—
98#340C2×C414yes—
99#340Q818yes—
100#336—22no—
101#336—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8C22×C412yes—
125C22×C8C22×C412yes—
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C22×C8C2×Q814yes—
129C22×C8#4112yes—
130C22×C8#4112yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C2×Q814yes—
165C2×C4⋊C4C2×Q814yes—
166C2×C4⋊C4#4114yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—82no—
180C2×C42C22×C412yes—
181C2×C42C2×D412yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184#107—28no—
185#106—44no—
186C4⋊C8—28no—
187#97—28no—
188#97—44no—
189C4×C8—44no—
190#95—28no—
191#95—24no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24C22×D411yescharacteristic
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C4×D412yes—
249C22×C4C4×D412yes—
250C22×C4#12114yes—
251C22×C4#13714yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×Q812yes—
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4H(5,𝔽2)12yes—
261C22×C42−1+412yes—
262C2×C8—24no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8C4×D414yes—
281C2×C8C4⋊Q814yes—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—48no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C4⋊C4—88no—
301C4⋊C4C4×Q818yes—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—24no—
309C42—41no—
310C42—41no—
311C42—42no—
312C42—88no—
313C23C2×C4×D411yescharacteristic
314C23C2×#12111yescharacteristic
315C23#53211yescharacteristic
316C23#54612yes—
317C23D4×Q812yes—
318C23C2×#13712yes—
319C23#57512yes—
320C23#57612yes—
321C23#57712yes—
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4C2×C4×D411yescharacteristic
375C2×C4C2×C4×Q812yes—
376C2×C4C4×#4111yescharacteristic
377C2×C4#52012yes—
378C2×C4#52511yescharacteristic
379C2×C4#52911yescharacteristic
380C2×C4C2×C4⋊Q811yescharacteristic
381C2×C4#53211yescharacteristic
382C2×C4#53312yes—
383C2×C4#53411yescharacteristic
384C2×C4#53611yescharacteristic
385C2×C4#53912yes—
386C2×C4#55212yes—
387C2×C4#55212yes—
388C2×C4#55312yes—
389C2×C4#55412yes—
390C2×C4#55412yes—
391C2×C4#55712yes—
392C8—24no—
393C8—24no—
394C8—42no—
395C8—42no—
396C22C4×#12111yescharacteristic
397C22#236012yes—
398C22#254311yescharacteristic
399C22#254611yescharacteristic
400C22#255612yes—
401C22C4×#13712yes—
402C22#294811yescharacteristic
403C22#295211yescharacteristic
404C22#305612yes—
405C22#306212yes—
406C22#317912yes—
407C22#318311yescharacteristic
408C22#318511yescharacteristic
409C22#331412yes—
410C22#331712yes—
411C22#331912yes—
412C22#332012yes—
413C22#338512yes—
414C22#338712yes—
415C22#338812yes—
416C22#338912yes—
417C4—42no—
418C4—42no—
419C4—44no—
420C4—44no—
421C4—48no—
422C4—84no—
423C4—84no—
424C4—162no—
425C4C4×C4⋊Q811yescharacteristic
426C4#232211yescharacteristic
427C4#237012yes—
428C4#258311yescharacteristic
429C4#258512yes—
430C4#261111yescharacteristic
431C2#1422611yescharacteristic
432C2#2152911yescharacteristic
433C2#2153911yescharacteristic
434C2#2297612yes—
435C2#2297912yes—
436C2#2856012yes—
437C2#2868612yes—
438C2#2897212yes—
439C2#2897312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #2086C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20862
C2×#19432
C2×#17712
#140452
#113772
#113272
#112312
#108842
#108812
#108802
#107032