Finite Group Databasebeta

← #276993

Subgroups — #276993

Order 512.

645 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4C4×#477C211yesmaximal, characteristic
5C2×#1676C212yesmaximal
6#10593C212yesmaximal
7#10543C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9216C212yesmaximal
11#8587C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#477C2212yes—
14C2×#477C2211yescharacteristic
15C2×C8×Q8C2212yes—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2083—22no—
20#1990C2212yes—
21#1987C2211yescharacteristic
22#1987C2212yes—
23#1947—21no—
24#1947—21no—
25#1947—21no—
26#1947—21no—
27#1947C2211yescharacteristic
28#1939—22no—
29#1939C2212yes—
30#1921—21no—
31#1863—21no—
32#1842—21no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1803—21no—
36#1803—21no—
37#1803C2211yescharacteristic
38C4×#107—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43#1676—28no—
44C2×#335C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#333C2212yes—
48#1296C2212yes—
49#1277C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#500—24no—
59#500—42no—
60#477—22no—
61#477—24no—
62#477—24no—
63#477D414yes—
64C8×Q8—24no—
65C8×Q8—42no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×#103—21no—
84C2×#103—22no—
85C2×#103—21no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—21no—
90#389—21no—
91#389—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2311yescharacteristic
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C43—21no—
106C43C2311yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#336—22no—
116#336—21no—
117#336D412yes—
118#335—24no—
119#333—24no—
120#333—24no—
121#333—42no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8C2×D411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140C4⋊Q8C2×D412yes—
141C4×Q8—22no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—82no—
147C4×Q8Q1614yes—
148C4×Q8C2×D412yes—
149C4×Q8#4112yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—41no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182#107—24no—
183#107—44no—
184#107—44no—
185#107—42no—
186#107—44no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—44no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—81no—
200C4⋊C8—82no—
201C4⋊C8Q1614yes—
202C4⋊C8C2×D412yes—
203#103—24no—
204#103—44no—
205#103—42no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—44no—
209C4×C8—42no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—41no—
214#95—41no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8#12112yes—
227C2×Q8C2×Q1612yes—
228C2×Q8#13712yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8C2×Q1614yes—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—81no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4#12112yes—
288C4⋊C4C2×Q1612yes—
289C42—22no—
290C42—22no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—24no—
296C42—22no—
297C42—41no—
298C42—42no—
299C42—41no—
300C42—41no—
301C42—42no—
302C42—81no—
303C42—82no—
304C42C2×Q1612yes—
305C42C2×Q1612yes—
306C42#13512yes—
307C42#13712yes—
308C42C22×D411yescharacteristic
309C42C2×#4111yescharacteristic
310C42H(5,𝔽2)11yescharacteristic
311C422−1+411yescharacteristic
312C23C2×#12111yescharacteristic
313Q8—22no—
314Q8—22no—
315Q8—22no—
316Q8—82no—
317Q8—82no—
318Q8—162no—
319Q8—162no—
320Q8#46214yes—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#46212yes—
352C2×C4#46212yes—
353C2×C4#46712yes—
354C2×C4#46712yes—
355C2×C4#47212yes—
356C2×C4C2×#12111yescharacteristic
357C2×C4#53711yescharacteristic
358C2×C4D4211yescharacteristic
359C2×C4#54711yescharacteristic
360C2×C4#54811yescharacteristic
361C2×C4#55311yescharacteristic
362C2×C4C22×Q1611yescharacteristic
363C2×C4C22×Q1611yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4#57511yescharacteristic
368C2×C4#57711yescharacteristic
369C2×C4#57811yescharacteristic
370C8—24no—
371C8—42no—
372C8—42no—
373C8—81no—
374C8—82no—
375C8—82no—
376C22#241611yescharacteristic
377C22C2×#46211yescharacteristic
378C22#304611yescharacteristic
379C22C2×#46711yescharacteristic
380C22#305611yescharacteristic
381C22C2×#47211yescharacteristic
382C22#306811yescharacteristic
383C4—24no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—82no—
393C4—161no—
394C4—162no—
395C4#167614yescomplemented
396C4C2×#46211yescharacteristic
397C4#303511yescharacteristic
398C4#324811yescharacteristic
399C4#324911yescharacteristic
400C4#327811yescharacteristic
401C4D4×Q1611yescharacteristic
402C4#336311yescharacteristic
403C4#337411yescharacteristic
404C2C2×#167611yescharacteristic
405C2#1917311yescharacteristic
406C2#1941711yescharacteristic
407C2#1950311yescharacteristic
408C2#2473211yescharacteristic
409C2#2475211yescharacteristic
410C2#2477011yescharacteristic
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
395. C4#1676semidirect

Maximal subgroups

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

subgroupindex
#331982
#286892
Q8×C4⋊C82
C4×#4772
C2×#16762
#105932
#105432
#96912
#96012
#92162
#85872