Finite Group Databasebeta

← #378943

Subgroups — #378943

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C211yesmaximal, characteristic
2#21159C212yesmaximal
3C2×#1941C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11330C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#10913C212yesmaximal
9#10896C212yesmaximal
10#10896C212yesmaximal
11C4×#341C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×C4×M16C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2296C2211yescharacteristic
22C2×#389C2212yes—
23#2090C418yescomplemented
24#1944—22no—
25#1943—22no—
26#1942—22no—
27#1941—22no—
28#1941—22no—
29#1941—22no—
30#1771—22no—
31C8×C4⋊C4—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1728C2211yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56#428—24no—
57#428—24no—
58#428—42no—
59#428C2×C418yes—
60C2×#107—22no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—42no—
68C2×#106C2×C414yes—
69#424—22no—
70#424C2×C414yes—
71C2×C4⋊C8—22no—
72#405—22no—
73#405—22no—
74#405—42no—
75C4×M16—24no—
76C4×M16—42no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—22no—
83C2×C4×C8—42no—
84C2×C4×C8D412yes—
85C2×C4×C8D412yes—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389D412yes—
91#389D412yes—
92#385—22no—
93C4×C4⋊C4—22no—
94C2×#95—22no—
95#343—24no—
96#343—24no—
97#343—42no—
98#343C2×C418yes—
99#341—22no—
100#341C2×C414yes—
101#341Q818yes—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—22no—
109#336—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4#4112yes—
113C23×C4C2411yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16C22×C412yes—
129C2×M16C22×C414yes—
130C2×M16C2×Q814yes—
131C2×M16C2×Q814yes—
132C2×M16#4112yes—
133C2×M16#4114yes—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8C22×C412yes—
144C22×C8C2×D412yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×Q814yes—
148C22×C8#4112yes—
149#117—21no—
150#117—21no—
151#117—22no—
152#117—22no—
153#117C22×C412yes—
154#117#4112yes—
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—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C22×C414yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42C2×D412yes—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183#107—28no—
184#107—44no—
185#106—28no—
186#106—28no—
187#106—44no—
188#106—44no—
189C4⋊C8—28no—
190#97—24no—
191#97—44no—
192#97—42no—
193C4×C8—24no—
194C4×C8—42no—
195C4×C8—42no—
196C4×C8—44no—
197C4×C8—42no—
198#95—22no—
199#95—22no—
200#95—42no—
201#95—42no—
202C24C2×#4111yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4#12112yes—
220C22×C4#12112yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C22×Q812yes—
225C22×C4C2×#4111yescharacteristic
226C22×C4C2×#4111yescharacteristic
227C22×C4C2×#4112yes—
228C22×C4C2×#4111yescharacteristic
229C22×C4H(5,𝔽2)12yes—
230C22×C42−1+412yes—
231M16—24no—
232M16—24no—
233M16—24no—
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16C4×Q818yes—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8C4×D414yes—
266C2×C8C4⋊Q812yes—
267C2×C8C4⋊Q812yes—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4C4×D414yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—41no—
294C42—41no—
295C23—22no—
296C23—22no—
297C23C4×#4111yescharacteristic
298C23C2×#12111yescharacteristic
299C23#52911yescharacteristic
300C23#55312yes—
301C23#55712yes—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—162no—
324C2×C4—162no—
325C2×C4#47114yes—
326C2×C4C2×C4×D411yescharacteristic
327C2×C4C2×C4×D411yescharacteristic
328C2×C4C2×C4×Q812yes—
329C2×C4#52012yes—
330C2×C4#52511yescharacteristic
331C2×C4C2×C4⋊Q811yescharacteristic
332C2×C4#53211yescharacteristic
333C2×C4#53211yescharacteristic
334C2×C4#53312yes—
335C2×C4#53411yescharacteristic
336C2×C4#53611yescharacteristic
337C2×C4#53912yes—
338C2×C4#54612yes—
339C2×C4D4×Q812yes—
340C2×C4#55212yes—
341C2×C4#55212yes—
342C2×C4#55412yes—
343C2×C4#55412yes—
344C8—24no—
345C8—22no—
346C8—22no—
347C8—28no—
348C8—42no—
349C8—42no—
350C8—44no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22C4×#12111yescharacteristic
358C22#237012yes—
359C22#254311yescharacteristic
360C22#254611yescharacteristic
361C22#258512yes—
362C22C2×#47112yes—
363C22#306712yes—
364C4—24no—
365C4—44no—
366C4—42no—
367C4—84no—
368C4—82no—
369C4—162no—
370C4C4×C4⋊Q811yescharacteristic
371C4#232211yescharacteristic
372C4#236012yes—
373C4#255612yes—
374C4#258311yescharacteristic
375C4#261111yescharacteristic
376C2—22no—
377C2—22no—
378C2#1422611yescharacteristic
379C2C4×#47112yes—
380C2#2859312yes—
381C2#2870011yescharacteristic
382C2#2870211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #2090C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20902
#211592
C2×#19412
#204352
#113662
#113302
#112382
#109132
#108962
C4×#3412