Finite Group Databasebeta

← #480311

Subgroups — #480311

Order 512.

965 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34249C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11520C214yesmaximal
5#11518C214yesmaximal
6#11454C212yesmaximal
7#11452C212yesmaximal
8C2×#519C411yescharacteristic
9C2×#519C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic
11#2983—21no—
12#2983—21no—
13#2351—21no—
14#2351—22no—
15#2351—21no—
16Q8×C4⋊C4—21no—
17Q8×C4⋊C4—22no—
18Q8×C4⋊C4—21no—
19#2317C412yes—
20#2317C411yescharacteristic
21#2317C2212yes—
22#2317C2211yescharacteristic
23#2296C412yes—
24#2296C411yescharacteristic
25#2296C2211yescharacteristic
26#2296C2212yes—
27#1467—22no—
28#1467—22no—
29#1464C2214yes—
30#1464C2214yes—
31#1461—21no—
32#1461—21no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2218yes—
37#1326C2214yes—
38C23×Q8C2×C411yescharacteristic
39#519—24no—
40#519—24no—
41#519—44no—
42C2×C4×Q8—22no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—44no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×#117C2×C412yes—
56C2×#117C2×C412yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C412yes—
60C2×#117C2×C411yescharacteristic
61C2×#117C2311yescharacteristic
62C2×#117C2312yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2×C411yescharacteristic
67#446—44no—
68#446—44no—
69#432—24no—
70#432—24no—
71#431—22no—
72#431—22no—
73#389—22no—
74#389—21no—
75#389—22no—
76#389—21no—
77#389—21no—
78#389—22no—
79#389—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—21no—
85#389—22no—
86#389—21no—
87#389—22no—
88#385C2×C411yescharacteristic
89#385C2×C412yes—
90#385C2×C412yes—
91#385C2×C412yes—
92#385C2×C412yes—
93#385C2×C411yescharacteristic
94#385C2×C412yes—
95#385C2311yescharacteristic
96#385C2312yes—
97#385C2311yescharacteristic
98#384—22no—
99#384—21no—
100#384—22no—
101#384—21no—
102#382—22no—
103#382—21no—
104#382—22no—
105#382—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113#324—22no—
114#324—22no—
115#324—24no—
116#324—24no—
117#323C2314yes—
118#323C2314yes—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22⋊C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C4×Q8—216no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—48no—
141C4×Q8—44no—
142C4×Q8—44no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—42no—
150#117—44no—
151#117—44no—
152#117—42no—
153#117M1614yes—
154#117M1614yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4C22×C411yescharacteristic
184C2×C22⋊C4C22×C412yes—
185C2×C22⋊C4C22×C411yescharacteristic
186C2×C22⋊C4C22×C411yescharacteristic
187C2×C22⋊C4C22×C412yes—
188C2×C22⋊C4C2411yescharacteristic
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212#97—44no—
213#97—44no—
214C4×C8—42no—
215C4×C8—42no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226C24C23×C411yescharacteristic
227C2×Q8—216no—
228C2×Q8—216no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8#9914yes—
237C2×Q8#10114yes—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—81no—
255C22×C4—82no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×M1612yes—
265C22×C4C2×M1612yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—44no—
269C2×C8—44no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—81no—
293C4⋊C4—82no—
294C4⋊C4—81no—
295C4⋊C4—82no—
296C22⋊C4—21no—
297C22⋊C4—22no—
298C22⋊C4—21no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4C2×M1612yes—
303C22⋊C4C2×M1614yes—
304C22⋊C4C2×M1612yes—
305C42—28no—
306C42—28no—
307C42—24no—
308C42—24no—
309C42—22no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—44no—
318C42—44no—
319C42—42no—
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23C22×C22⋊C411yescharacteristic
325C23C22×M1611yescharacteristic
326C23C22×M1611yescharacteristic
327Q8—216no—
328Q8—28no—
329Q8—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4#40712yes—
375C2×C4#40712yes—
376C2×C4C2×#9914yes—
377C2×C4C2×#9912yes—
378C2×C4C2×#10114yes—
379C2×C4C2×#10112yes—
380C8—84no—
381C8—84no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22C2×#40711yescharacteristic
392C22C22×#9911yescharacteristic
393C22C22×#10111yescharacteristic
394C4—22no—
395C4—24no—
396C4—24no—
397C4—44no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—82no—
408C4—81no—
409C4#146314yes—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—41no—
414C2C2×#146311yescharacteristic
415C2#3334911yescharacteristic
416C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutator, prime radicalC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#342492
#173712
#115202
#115182
#114542
#114522