Finite Group Databasebeta

← #224473

Subgroups — #224473

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic
3#22722C211yesmaximal, characteristic
4#9907C212yesmaximal
5#9904C212yesmaximal
6#9898C211yesmaximal, characteristic
7#9896C211yesmaximal, characteristic
8#8590C212yesmaximal
9#8587C211yesmaximal, characteristic
10#8232C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#446—21no—
13#2868C2211yescharacteristic
14#2859C2212yes—
15#2582C412yes—
16#2567—21no—
17#2552—21no—
18#2550—22no—
19#2507—22no—
20#2451C414yes—
21#2369—21no—
22#2355—21no—
23Q8×C4⋊C4C412yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
26C42×Q8C2211yescharacteristic
27#1984—21no—
28#1940—22no—
29#1939—21no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2212yes—
34#1831C2212yes—
35#1770—21no—
36#1767C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#333C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2211yescharacteristic
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×#125—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60#446—44no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×#97—22no—
74C2×#97—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#398—24no—
78#398—24no—
79#395—24no—
80#391—22no—
81#391—24no—
82#391C2×C414yes—
83#391C2×C412yes—
84#389—22no—
85#389—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—24no—
92#389C2×C414yes—
93#389C2×C414yes—
94#387—24no—
95#384—22no—
96#384—21no—
97#384—21no—
98#384C2×C412yes—
99#384C2311yescharacteristic
100#384C2312yes—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D411yescharacteristic
113C43C2312yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—21no—
118#336—22no—
119#336—22no—
120#336—21no—
121#333—22no—
122#333—24no—
123#333Q814yescomplemented
124C22×Q8—22no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22×C412yes—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—81no—
138C4⋊Q8—81no—
139C4⋊Q8—82no—
140C4⋊Q8C2×Q812yes—
141C4⋊Q8#4112yes—
142#125—82no—
143#125—82no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—44no—
150C4×Q8—84no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C414yes—
171C2×C4⋊C4C22×C412yes—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2411yescharacteristic
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—44no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—81no—
201C4⋊C8—82no—
202C4⋊C8—81no—
203C4⋊C8C2×Q812yes—
204C4⋊C8C2×Q812yes—
205#97—44no—
206#97—82no—
207C4×C8—42no—
208C4×C8—42no—
209#95—24no—
210#95—22no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—24no—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8C4×Q814yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—44no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—21no—
294C42—22no—
295C42—22no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42—42no—
302C42—41no—
303C42—44no—
304C42—84no—
305C42AGL(1,ℤ/8)12yes—
306C42#13712yes—
307C42#13712yes—
308C42#13712yes—
309C42C22×Q811yescharacteristic
310C42C2×#4111yescharacteristic
311C422−1+411yescharacteristic
312C422−1+411yescharacteristic
313C23C22×C22⋊C411yescharacteristic
314Q8—44no—
315Q8—44no—
316Q8—44no—
317Q8—164no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4#41312yes—
349C2×C4#41712yes—
350C2×C4#41712yes—
351C2×C4#41912yes—
352C2×C4#41912yes—
353C2×C4C2×C4×Q811yescharacteristic
354C2×C4#51911yescharacteristic
355C2×C4#54811yescharacteristic
356C2×C4#54811yescharacteristic
357C2×C4D4×Q811yescharacteristic
358C2×C4D4×Q811yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C2×#13711yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4C2×#13711yescharacteristic
363C2×C4#57611yescharacteristic
364C2×C4#57711yescharacteristic
365C2×C4#57711yescharacteristic
366C2×C4#57811yescharacteristic
367C8—42no—
368C8—42no—
369C8—81no—
370C8—82no—
371C8—81no—
372C8—82no—
373C22Q8×C22⋊C411yescharacteristic
374C22C2×#41311yescharacteristic
375C22#288811yescharacteristic
376C22C2×#41911yescharacteristic
377C22C2×#41711yescharacteristic
378C22#289711yescharacteristic
379C22#289811yescharacteristic
380C4—22no—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—41no—
388C4—42no—
389C4—44no—
390C4—84no—
391C4—82no—
392C4#155914yes—
393C4#336811yescharacteristic
394C4#336911yescharacteristic
395C4#337411yescharacteristic
396C4#337511yescharacteristic
397C4#338511yescharacteristic
398C4#338611yescharacteristic
399C4#338711yescharacteristic
400C4#338711yescharacteristic
401C2C2×#155911yescharacteristic
402C2#1776011yescharacteristic
403C2#1776211yescharacteristic
404C2#1778111yescharacteristic
405C2#2251511yescharacteristic
406C2#2252411yescharacteristic
407C2#2252511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
123. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#228252
#227222
#99072
#99042
#98982
#98962
#85902
#85872
#82322
C4×#3332