Finite Group Databasebeta

← #224207

Subgroups — #224207

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3#22821C211yesmaximal, characteristic
4#9901C212yesmaximal
5#9901C212yesmaximal
6#9898C211yesmaximal, characteristic
7#9896C211yesmaximal, characteristic
8#8593C212yesmaximal
9#8586C211yesmaximal, characteristic
10C4×#332C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2859C2212yes—
14#2582—21no—
15#2567—21no—
16#2552C412yes—
17#2550C414yes—
18#2507—22no—
19#2451—22no—
20#2369—21no—
21#2355C412yes—
22Q8×C4⋊C4—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
25C42×Q8C2211yescharacteristic
26#1984—21no—
27#1984—21no—
28#1916—22no—
29#1916—22no—
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1831C2212yes—
33#1831C2212yes—
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40C2×#332C2212yes—
41C2×#332C2211yescharacteristic
42C2×#332C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×#125C2×C414yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#398—24no—
75#398—24no—
76#395—24no—
77#391—22no—
78#391—22no—
79#391—21no—
80#391—24no—
81#389—22no—
82#389—22no—
83#389—21no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389C2×C412yes—
90#389C2×C414yes—
91#387—24no—
92#384—22no—
93#384—21no—
94#384—21no—
95#384C2×C412yes—
96#384C2311yescharacteristic
97#384C2312yes—
98#382C2×C414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C43C2312yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#336—22no—
119#336—22no—
120#332—24no—
121#332—22no—
122#332Q814yescomplemented
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8C22⋊C411yescharacteristic
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—81no—
137C4⋊Q8—81no—
138C4⋊Q8—82no—
139C4⋊Q8C2×Q812yes—
140C4⋊Q8#4112yes—
141#125—84no—
142C4×Q8—24no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—82no—
150C4×Q8—82no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C414yes—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
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—21no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—81no—
200C4⋊C8—81no—
201C4⋊C8C2×Q812yes—
202C4⋊C8C2×Q812yes—
203C4×C8—42no—
204C4×C8—42no—
205C4×C8—82no—
206C4×C8—82no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—24no—
212#95—24no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
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—41no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4#10114yes—
286C4⋊C4C4×Q814yes—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—22no—
294C42—21no—
295C42—22no—
296C42—21no—
297C42—22no—
298C42—21no—
299C42—22no—
300C42—24no—
301C42—42no—
302C42—41no—
303C42—44no—
304C42—84no—
305C42#13512yes—
306C42#13512yes—
307C42#13512yes—
308C42#13512yes—
309C42C22×Q811yescharacteristic
310C42C2×#4111yescharacteristic
311C422−1+411yescharacteristic
312C422−1+411yescharacteristic
313C23C22×C22⋊C411yescharacteristic
314Q8—44no—
315Q8—42no—
316Q8—42no—
317Q8—42no—
318Q8—42no—
319Q8—164no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4C2×#10114yes—
350C2×C4C2×#10112yes—
351C2×C4#41612yes—
352C2×C4#41612yes—
353C2×C4#41612yes—
354C2×C4#41612yes—
355C2×C4C2×C4×Q811yescharacteristic
356C2×C4#51911yescharacteristic
357C2×C4#54811yescharacteristic
358C2×C4#54811yescharacteristic
359C2×C4D4×Q811yescharacteristic
360C2×C4D4×Q811yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4#57611yescharacteristic
366C2×C4#57711yescharacteristic
367C2×C4#57711yescharacteristic
368C2×C4#57811yescharacteristic
369C8—42no—
370C8—42no—
371C8—81no—
372C8—82no—
373C8—81no—
374C8—82no—
375C22Q8×C22⋊C411yescharacteristic
376C22C22×#10111yescharacteristic
377C22#288811yescharacteristic
378C22C2×#41611yescharacteristic
379C22C2×#41611yescharacteristic
380C22#289711yescharacteristic
381C22#289811yescharacteristic
382C4—22no—
383C4—22no—
384C4—24no—
385C4—21no—
386C4—21no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—41no—
391C4—44no—
392C4—84no—
393C4—82no—
394C4#154414yes—
395C4#336011yescharacteristic
396C4#336111yescharacteristic
397C4#336411yescharacteristic
398C4#336511yescharacteristic
399C4#338011yescharacteristic
400C4#338111yescharacteristic
401C4#338111yescharacteristic
402C4#338311yescharacteristic
403C2C2×#154411yescharacteristic
404C2#1772411yescharacteristic
405C2#1772611yescharacteristic
406C2#1777411yescharacteristic
407C2Q8×#10111yescharacteristic
408C2#2251911yescharacteristic
409C2#2252111yescharacteristic
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
122. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#228212
#99012
#98982
#98962
#85932
#85862
C4×#3322