Finite Group Databasebeta

← #58592

Subgroups — #58592

Order 256.

697 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2839C212yesmaximal
2#2833C211yesmaximal, characteristic
3#2809C211yesmaximal, characteristic
4#2799C211yesmaximal, characteristic
5#2766C212yesmaximal
6#2755C212yesmaximal
7#2726C212yesmaximal
8#2726C212yesmaximal
9#2719C212yesmaximal
10#2713C212yesmaximal
11#2713C212yesmaximal
12#2655C212yesmaximal
13#2634C211yesmaximal, characteristic
14#2586C211yesmaximal, characteristic
15#2585C212yesmaximal
16#2581C211yesmaximal, characteristic
17#2550C212yesmaximal
18#2546C211yesmaximal, characteristic
19#2453C212yesmaximal
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#125C2211yescharacteristic
24C2×#125C2212yes—
25C2×#125C2212yes—
26C2×#125C2212yes—
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29#401C2212yes—
30#400C2211yescharacteristic
31#400C2212yes—
32#400C2212yes—
33#400C2211yescharacteristic
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#400C2212yes—
38#400C2212yes—
39#400C2211yescharacteristic
40#400C2211yescharacteristic
41#400C2212yes—
42#400C2212yes—
43#398C2211yescharacteristic
44#398C2212yes—
45#398C2212yes—
46#398C2212yes—
47#398C2212yes—
48#398C2212yes—
49#398C2212yes—
50#398C2212yes—
51#398C2212yes—
52#398C2212yes—
53#398C2211yescharacteristic
54#398C2212yes—
55#398C2212yes—
56#398C2212yes—
57#398C2212yes—
58#398C2212yes—
59#398C2212yes—
60#395C2212yes—
61#395C2211yescharacteristic
62#395C2211yescharacteristic
63#395C2212yes—
64#395C2212yes—
65#391C2211yescharacteristic
66#391C2212yes—
67#391C2212yes—
68#391C2211yescharacteristic
69#391C2212yes—
70#391C2212yes—
71#391C2212yes—
72#389C2212yes—
73#389C2212yes—
74#389C2212yes—
75#389C2212yes—
76#389C2212yes—
77#389C2212yes—
78#389C2212yes—
79#389C2212yes—
80#389C2212yes—
81#389C2212yes—
82#389C2212yes—
83#389C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#387C2212yes—
92#387C2212yes—
93#387C2212yes—
94#387C2212yes—
95#387C2212yes—
96#387C2212yes—
97#387C2212yes—
98#387C2212yes—
99#387C2212yes—
100#387C2212yes—
101#383C2211yescharacteristic
102#382C2211yescharacteristic
103#382C2212yes—
104C4×C4⋊C4C2211yescharacteristic
105C4×C4⋊C4C2212yes—
106C4×C4⋊C4C2212yes—
107C22×Q8C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C22×Q8C2311yescharacteristic
110C22×Q8C2311yescharacteristic
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114#125—42no—
115#125—44no—
116#125—44no—
117#125—44no—
118C4×Q8—42no—
119C4×Q8—42no—
120C2×C4⋊C4C2312yes—
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2311yescharacteristic
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C42C2311yescharacteristic
156C2×C42C2311yescharacteristic
157C2×C42C2311yescharacteristic
158C2×C42C2312yes—
159C2×C42C2312yes—
160C2×C42C2312yes—
161C2×C42C2312yes—
162C2×C42C2312yes—
163C2×C42C2311yescharacteristic
164C2×C42C2311yescharacteristic
165#95C2312yes—
166#95C2312yes—
167#95C2312yes—
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2312yes—
174#95C2311yescharacteristic
175#95C2311yescharacteristic
176#95C2312yes—
177#95C2312yes—
178#95C2311yescharacteristic
179#95C2311yescharacteristic
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203C2×Q8—22no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C22×C4C2412yes—
215C22×C4C2411yescharacteristic
216C22×C4C2411yescharacteristic
217C22×C4C2411yescharacteristic
218C22×C4C2412yes—
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2412yes—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2411yescharacteristic
225C22×C4C2412yes—
226C22×C4C2412yes—
227C22×C4C2412yes—
228C22×C4C2411yescharacteristic
229C22×C4C2411yescharacteristic
230C22×C4C2412yes—
231C22×C4C2411yescharacteristic
232C22×C4C2412yes—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—24no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—41no—
277C42—41no—
278C23C2511yescharacteristic
279Q8—44no—
280Q8—44no—
281Q8—82no—
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4H(5,𝔽2)12yes—
332C2×C4H(5,𝔽2)12yes—
333C2×C42−1+414yes—
334C2×C42−1+414yes—
335C2×C42−1+412yes—
336C22C2×H(5,𝔽2)11yescharacteristic
337C22C2×H(5,𝔽2)11yescharacteristic
338C22C2×2−1+411yescharacteristic
339C22C2×2−1+412yes—
340C22C2×2−1+412yes—
341C4—22no—
342C4—22no—
343C4—24no—
344C4—22no—
345C4—24no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—42no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—42no—
360C2#353611yescharacteristic
361C2#353811yescharacteristic
362C2#354411yescharacteristic
363C2#355212yes—
364C2#355612yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#28392
#28332
#28092
#27992
#27662
#27552
#27262
#27192
#27132
#26552
#26342
#25862
#25852
#25812
#25502
#25462
#24532