Finite Group Databasebeta

← #17769

Subgroups — #17769

Order 256.

459 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic, complemented
3#2925C211yesmaximal, characteristic, complemented
4#1561C211yesmaximal, characteristic, complemented
5#1557C211yesmaximal, characteristic, complemented
6#1557C211yesmaximal, characteristic, complemented
7#1556C211yesmaximal, characteristic, complemented
8#1553C211yesmaximal, characteristic, complemented
9#1553C211yesmaximal, characteristic, complemented
10#1552C211yesmaximal, characteristic, complemented
11#1552C211yesmaximal, characteristic, complemented
12#1547C211yesmaximal, characteristic, complemented
13#1546C211yesmaximal, characteristic, complemented
14#1544C211yesmaximal, characteristic, complemented
15#1540C211yesmaximal, characteristic, complemented
16#556—21no—
17#555C412yes—
18#554C412yes—
19#553C412yes—
20#553C412yes—
21#548—21no—
22#548—21no—
23#547—21no—
24#542—21no—
25#542—21no—
26#537—21no—
27#533C2211yescharacteristic
28#533C2211yescharacteristic
29#532C2211yescharacteristic
30#532C2211yescharacteristic
31#532C2211yescharacteristic
32C2×C4⋊Q8C2211yescharacteristic
33#520—21no—
34C4×#41C2211yescharacteristic
35#446—21no—
36#436—21no—
37#435—21no—
38#435—21no—
39#433—21no—
40#433—21no—
41#432C2211yescharacteristic
42#432C2211yescharacteristic
43#432C2211yescharacteristic
44#432C2211yescharacteristic
45#431C2211yescharacteristic
46#431C2211yescharacteristic
47#431C2211yescharacteristic
48#423C2211yescharacteristic
49#423C2211yescharacteristic
50#423C2211yescharacteristic
51C2×C4⋊C8C2211yescharacteristic
52C2×C4⋊C8C2211yescharacteristic
53#408—21no—
54#405—21no—
55#332C2211yescharacteristic
56#332C2211yescharacteristic
57#332C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#332C2211yescharacteristic
66#332C2211yescharacteristic
67#331C2211yescharacteristic
68#331C2211yescharacteristic
69#331C2211yescharacteristic
70#331C2211yescharacteristic
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—21no—
74C2×#41D411yescharacteristic
75C22×Q8—21no—
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2×C412yes—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127C2311yescharacteristic
88#126—22no—
89#126—22no—
90#125C2×C412yes—
91#125C2×C412yes—
92#124—22no—
93#124—22no—
94#124—21no—
95#124—21no—
96#124—21no—
97#123—22no—
98#123—22no—
99C22⋊Q8—22no—
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106#121—22no—
107#121—22no—
108#121—22no—
109#121—21no—
110#121—21no—
111#121—21no—
112#121C2×C412yes—
113#121C2×C412yes—
114#121C2×C412yes—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8C2×C411yescharacteristic
120C4×Q8D411yescharacteristic
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4C2×C412yes—
127C4×D4C2×C412yes—
128C4×D4C2×C412yes—
129C4×D4C2×C411yescharacteristic
130C4×D4C2×C411yescharacteristic
131C4×D4C2×C411yescharacteristic
132C4×D4D411yescharacteristic
133C4×D4D411yescharacteristic
134C4×D4D411yescharacteristic
135#117—22no—
136#117C2×C412yes—
137#117C2×C412yes—
138#117D411yescharacteristic
139#117D411yescharacteristic
140#117D411yescharacteristic
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×C412yes—
144C2×C42C2311yescharacteristic
145C2×C42C2311yescharacteristic
146C2×C42C2311yescharacteristic
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8C2311yescharacteristic
153C4⋊C8C2311yescharacteristic
154C4⋊C8C2311yescharacteristic
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167#97—21no—
168#97—21no—
169#97—21no—
170#97—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174#41—41no—
175#41—41no—
176#41—42no—
177#41—42no—
178#41—81no—
179#41—81no—
180#41—81no—
181#41—81no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8C22⋊C411yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4C22⋊C411yescharacteristic
195C2×D4C22⋊C411yescharacteristic
196C2×D4C22⋊C411yescharacteristic
197C2×D4C22×C412yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4C22×C411yescharacteristic
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211M16—42no—
212M16—81no—
213M16—81no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—42no—
223C2×C8—81no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4C22⋊C411yescharacteristic
234C4⋊C4C22⋊C411yescharacteristic
235C4⋊C4C22⋊C411yescharacteristic
236C4⋊C4C22⋊C411yescharacteristic
237C4⋊C4C22⋊C411yescharacteristic
238C4⋊C4C22⋊C411yescharacteristic
239C4⋊C4C22×C412yes—
240C4⋊C4C22×C412yes—
241C4⋊C4C22×C412yes—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4C22⋊C411yescharacteristic
251C22⋊C4C22⋊C411yescharacteristic
252C22⋊C4C22⋊C411yescharacteristic
253C22⋊C4C22⋊C411yescharacteristic
254C22⋊C4C22⋊C411yescharacteristic
255C22⋊C4C22⋊C411yescharacteristic
256C42—22no—
257C42—22no—
258C42C22×C411yescharacteristic
259C42C22×C411yescharacteristic
260C42C22×C411yescharacteristic
261C42C2×D411yescharacteristic
262C42C2×D411yescharacteristic
263C42C2×D411yescharacteristic
264C42C2×D411yescharacteristic
265C42C2×D411yescharacteristic
266C42C2×D411yescharacteristic
267C42C2411yescharacteristic
268C23—22no—
269C23C2×C22⋊C411yescharacteristic
270C23C2×C22⋊C411yescharacteristic
271C23C2×C22⋊C411yescharacteristic
272Q8—41no—
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—82no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—44no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—82no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C2×C22⋊C411yescharacteristic
314C2×C4C2×C22⋊C411yescharacteristic
315C2×C4C2×C22⋊C411yescharacteristic
316C2×C4C2×C22⋊C411yescharacteristic
317C2×C4C2×C22⋊C411yescharacteristic
318C2×C4C2×C22⋊C411yescharacteristic
319C2×C4C2×C22⋊C411yescharacteristic
320C2×C4C2×C22⋊C411yescharacteristic
321C2×C4C2×C22⋊C411yescharacteristic
322C2×C4C23×C411yescharacteristic
323C2×C4C22×D411yescharacteristic
324C2×C4C22×D411yescharacteristic
325C8—41no—
326C8—41no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C8—41no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22C22×C22⋊C411yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4#57511yescharacteristic
360C4#57511yescharacteristic
361C4#57611yescharacteristic
362C4#57711yescharacteristic
363C2—41no—
364C2—41no—
365C2—41no—
366C2—82no—
367C2#288911yescharacteristic
368C2#289611yescharacteristic
369C2#289711yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3511C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2989C2semidirect
3. #2925C2semidirect
4. #1561C2semidirect
5. #1557C2semidirect
6. #1557C2semidirect
7. #1556C2semidirect
8. #1553C2semidirect
9. #1553C2semidirect
10. #1552C2semidirect
11. #1552C2semidirect
12. #1547C2semidirect
13. #1546C2semidirect
14. #1544C2semidirect
15. #1540C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#29892
#29252
#15612
#15572
#15562
#15532
#15522
#15472
#15462
#15442
#15402