Finite Group Databasebeta

← #318175

Subgroups — #318175

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28284C211yesmaximal, characteristic
2#28255C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5C2×#1672C212yesmaximal
6#10482C212yesmaximal
7#10479C212yesmaximal
8#8805C211yesmaximal, characteristic
9#8643C212yesmaximal
10#8587C211yesmaximal, characteristic
11#8586C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#477C2212yes—
14C2×#462—21no—
15C2×#462C2212yes—
16C2×C8×Q8C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2068—22no—
21#2056—22no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1984C2212yes—
25#1981—21no—
26#1947—21no—
27#1941—21no—
28#1941—21no—
29#1939—21no—
30#1939—21no—
31#1844C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1803—21no—
36#1803C2212yes—
37#1803C2212yes—
38C4×#106—21no—
39C4×#106—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4—21no—
42C42×C8C2211yescharacteristic
43#1672—28no—
44C2×#334C2212yes—
45C2×#334C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50C22×Q16—22no—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2312yes—
57#498—22no—
58#498D414yes—
59#477—24no—
60#477—42no—
61#462—24no—
62#462—42no—
63#462—44no—
64C8×Q8—24no—
65C8×Q8—24no—
66C2×#107—22no—
67C2×#106—21no—
68C2×#106—21no—
69C2×#106—21no—
70C2×#106—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C4×C8—22no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391—21no—
93#391—21no—
94#391D412yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#384D411yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C43C2311yescharacteristic
111#340—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#340—22no—
116#340—22no—
117#336—22no—
118#336—22no—
119#334—24no—
120#334—42no—
121#333—24no—
122#332—24no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C2×D412yes—
126C2×Q16—44no—
127C2×Q16—44no—
128C2×Q16—82no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C4⋊Q8—22no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8SD1614yes—
144C4⋊Q8C2×D412yes—
145C4⋊Q8C2×D412yes—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—82no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×D412yes—
171C2×C42—22no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181#107—44no—
182#106—24no—
183#106—24no—
184#106—42no—
185#106—42no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—42no—
193C4⋊C8—44no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8—81no—
198C4⋊C8—81no—
199#103—44no—
200#103—44no—
201#103—44no—
202#103—82no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—41no—
208C4×C8—44no—
209C4×C8Q1614yescomplemented
210C4×C8C2×D412yes—
211C4×C8#4112yes—
212#95—22no—
213#95—22no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—81no—
232C2×Q8—81no—
233C2×Q8C2×SD1614yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C2×#4111yescharacteristic
247Q16—88no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8#12112yes—
265C2×C8#12112yes—
266C2×C8C2×Q1612yes—
267C2×C8C2×Q1612yes—
268C2×C8#13712yes—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—41no—
303C42—42no—
304C42—42no—
305C42—82no—
306C42—82no—
307C42C2×SD1612yes—
308C42C2×Q1612yes—
309C42#13512yes—
310C42#13712yes—
311C42C22×D411yescharacteristic
312C42C2×#4111yescharacteristic
313C42H(5,𝔽2)11yescharacteristic
314C422−1+411yescharacteristic
315C23C2×#12111yescharacteristic
316Q8—24no—
317Q8—24no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—42no—
322Q8—82no—
323Q8—82no—
324Q8—81no—
325Q8—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#46212yes—
354C2×C4#46212yes—
355C2×C4#46512yes—
356C2×C4#46512yes—
357C2×C4#47112yes—
358C2×C4C2×#12111yescharacteristic
359C2×C4#53711yescharacteristic
360C2×C4D4211yescharacteristic
361C2×C4#54711yescharacteristic
362C2×C4#54711yescharacteristic
363C2×C4#55511yescharacteristic
364C2×C4C22×SD1611yescharacteristic
365C2×C4C22×Q1611yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4#57511yescharacteristic
370C2×C4#57711yescharacteristic
371C2×C4#57711yescharacteristic
372C8—22no—
373C8—22no—
374C8—41no—
375C8—82no—
376C8—82no—
377C8—82no—
378C8#46214yescomplemented
379C22#253711yescharacteristic
380C22C2×#46211yescharacteristic
381C22#303711yescharacteristic
382C22C2×#46511yescharacteristic
383C22#306011yescharacteristic
384C22C2×#47111yescharacteristic
385C22#306711yescharacteristic
386C4—22no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—41no—
392C4—82no—
393C4—82no—
394C4—82no—
395C4—161no—
396C4—161no—
397C4#167214yescomplemented
398C4C2×#46211yescharacteristic
399C4#303711yescharacteristic
400C4#324911yescharacteristic
401C4#325511yescharacteristic
402C4D4×SD1611yescharacteristic
403C4#328411yescharacteristic
404C4#333511yescharacteristic
405C4#334011yescharacteristic
406C2C2×#167211yescharacteristic
407C2#1921011yescharacteristic
408C2#1926011yescharacteristic
409C2#1934311yescharacteristic
410C2#2822311yescharacteristic
411C2#2823911yescharacteristic
412C2#2825511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
209. C4×C8Q16semidirect
378. C8#462semidirect
397. C4#1672semidirect

Maximal subgroups

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

subgroupindex
#282842
#282552
C4×#4982
C8×C4⋊Q82
C2×#16722
#104822
#104792
#88052
#86432
#85872
#85862