Finite Group Databasebeta

← #61852

Subgroups — #61852

Order 256.

796 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3556C211yesmaximal, characteristic
2#3502C211yesmaximal, characteristic
3#3390C214yesmaximal, complemented
4#3387C212yesmaximal, complemented
5#3383C212yesmaximal, complemented
6#3374C212yesmaximal, complemented
7#3374C214yesmaximal, complemented
8#3363C212yesmaximal, complemented
9#3242C214yesmaximal
10#3237C211yesmaximal, characteristic
11#3231C211yesmaximal, characteristic
12#3127C214yesmaximal
13#3113C211yesmaximal, characteristic
14#3104C211yesmaximal, characteristic
15#2985C211yesmaximal, characteristic
16#564—22no—
17Q82—21no—
18Q82C2212yes—
19#557—21no—
20#557—21no—
21#557—21no—
22#557—22no—
23#557C2212yes—
24#557C2212yes—
25#557C2212yes—
26#557C2214yes—
27#557C2212yes—
28#557C2212yes—
29#544—22no—
30#544C2212yes—
31#541—22no—
32#541—22no—
33#541C2214yes—
34#533—22no—
35#533C2211yescharacteristic
36#533C2211yescharacteristic
37#533C2211yescharacteristic
38#533C2211yescharacteristic
39#519C2212yes—
40C2×C4×Q8C2211yescharacteristic
41#501C2214yes—
42#501C2212yes—
43#500C2211yescharacteristic
44#499C2214yes—
45#498C2211yescharacteristic
46#491C2212yes—
47#491C2214yes—
48#487C2211yescharacteristic
49#487C2214yes—
50#487C2211yescharacteristic
51#485C2214yes—
52#485C2214yes—
53#484C2212yes—
54#483C2212yes—
55#479C2214yes—
56#479C2214yes—
57#479C2214yes—
58#479C2214yes—
59#479C2214yes—
60#477C2212yes—
61#477C2212yes—
62#477C2214yes—
63#475C2212yes—
64#475C2212yes—
65#475C2214yes—
66#462C2212yes—
67#462C2212yes—
68#462C2214yes—
69#462C2214yes—
70#451C2212yes—
71#451C2212yes—
72#446C2212yes—
73#446C2214yes—
74C8×Q8C2212yes—
75#441C2214yes—
76#441C2214yes—
77#441C2214yes—
78C4×Q16C2212yes—
79C4×Q16C2212yes—
80C4×Q16C2214yes—
81C4×Q16C2214yes—
82#433C2214yes—
83#432C2211yescharacteristic
84#431C2211yescharacteristic
85C22×Q8C2311yescharacteristic
86C2×Q16C2312yes—
87C2×Q16C2312yes—
88C2×Q16C2314yes—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—24no—
93C4⋊Q8—22no—
94C4⋊Q8—24no—
95C4⋊Q8—22no—
96C4⋊Q8D411yescharacteristic
97C4⋊Q8D411yescharacteristic
98C4⋊Q8C2311yescharacteristic
99C4⋊Q8C2311yescharacteristic
100C4⋊Q8C2314yes—
101C4⋊Q8C2311yescharacteristic
102C4⋊Q8C2311yescharacteristic
103C4⋊Q8C2311yescharacteristic
104C4⋊Q8C2311yescharacteristic
105C4⋊Q8C2314yes—
106#126—24no—
107#126—24no—
108#125—24no—
109#125—22no—
110#125—22no—
111#125—24no—
112#125—24no—
113#125—24no—
114#125—24no—
115#125—24no—
116#125—24no—
117#125—24no—
118#125—24no—
119#125—22no—
120#125—21no—
121#125—21no—
122#125D412yes—
123#125C2312yes—
124#125C2314yes—
125#125C2314yes—
126C22⋊Q8—21no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130C22⋊Q8—24no—
131C22⋊Q8D412yes—
132C22⋊Q8D412yes—
133C22⋊Q8C2312yes—
134C22⋊Q8C2312yes—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—24no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8C2312yes—
151C4×Q8C2312yes—
152C4×Q8C2314yes—
153C4×Q8C2314yes—
154C4×Q8C2312yes—
155C4×Q8C2312yes—
156C4×Q8C2314yes—
157C4×Q8C2314yes—
158C4×Q8C2312yes—
159C4×Q8C2312yes—
160#117—21no—
161#117—21no—
162#117—24no—
163#117C2312yes—
164#117C2312yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C2311yescharacteristic
167C2×C42—21no—
168C2×C42C2311yescharacteristic
169#107C2312yes—
170#107C2314yes—
171#107C2312yes—
172#107C2314yes—
173#106C2314yes—
174#106C2314yes—
175#106C2314yes—
176C4⋊C8C2312yes—
177C4⋊C8C2312yes—
178C4⋊C8C2314yes—
179C4⋊C8C2314yes—
180#103C2312yes—
181#103C2314yes—
182#103C2312yes—
183#103C2312yes—
184#103C2314yes—
185#103C2314yes—
186#103C2312yes—
187#103C2314yes—
188#103C2314yes—
189#103C2314yes—
190C22⋊C8C2312yes—
191C22⋊C8C2312yes—
192#97C2312yes—
193#97C2314yes—
194C4×C8C2311yescharacteristic
195C4×C8C2311yescharacteristic
196C4×C8C2314yes—
197C2×Q8—24no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—24no—
204C2×Q8C2×D412yes—
205C2×Q8C2×D412yes—
206C2×Q8C2412yes—
207C2×Q8C2412yes—
208C2×Q8C2412yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2411yescharacteristic
214Q16—24no—
215Q16—24no—
216Q16—28no—
217C2×C8C2412yes—
218C2×C8C2412yes—
219C2×C8C2414yes—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4C2×D412yes—
249C4⋊C4C2×D412yes—
250C4⋊C4C2×D412yes—
251C4⋊C4C2×D414yes—
252C4⋊C4C2×D412yes—
253C4⋊C4C2×D414yes—
254C4⋊C4C2412yes—
255C4⋊C4C2412yes—
256C4⋊C4C2412yes—
257C4⋊C4C2412yes—
258C4⋊C4C2412yes—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4C2×D414yes—
263C42—22no—
264C42—22no—
265C42—24no—
266C42—24no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42—24no—
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2411yescharacteristic
275C42C2414yes—
276C42C2411yescharacteristic
277C23C22×D411yescharacteristic
278Q8—24no—
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—22no—
283Q8—24no—
284Q8—42no—
285Q8—42no—
286Q8—42no—
287Q8—42no—
288Q8—81no—
289Q8—84no—
290Q82−1+414yes—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C22×D411yescharacteristic
307C2×C4C22×D414yes—
308C2×C4C22×D414yes—
309C2×C4C22×D411yescharacteristic
310C2×C4C22×D411yescharacteristic
311C2×C4C2511yescharacteristic
312C8—22no—
313C8—22no—
314C8—24no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22C23×D411yescharacteristic
319C4—21no—
320C4—21no—
321C4—24no—
322C4—24no—
323C4—21no—
324C4—21no—
325C4—24no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4#57812yes—
334C4C2×2−1+412yes—
335C2—41no—
336C2#348611yescharacteristic
337C2C2×#57811yescharacteristic
338C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC23C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #3390C2semidirect
4. #3387C2semidirect
5. #3383C2semidirect
6. #3374C2semidirect
7. #3374C2semidirect
8. #3363C2semidirect

Maximal subgroups

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

subgroupindex
#35562
#35022
#33902
#33872
#33832
#33742
#33632
#32422
#32372
#32312
#31272
#31132
#31042
#29852