Finite Group Databasebeta

← #17766

Subgroups — #17766

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic, complemented
3#2925C211yesmaximal, characteristic, complemented
4#1560C211yesmaximal, characteristic, complemented
5#1559C211yesmaximal, characteristic, complemented
6#1557C211yesmaximal, characteristic, complemented
7#1555C211yesmaximal, characteristic, complemented
8#1554C211yesmaximal, characteristic, complemented
9#1553C211yesmaximal, characteristic, complemented
10#1551C211yesmaximal, characteristic, complemented
11#1551C211yesmaximal, characteristic, complemented
12#1547C211yesmaximal, characteristic, complemented
13#1545C211yesmaximal, characteristic, complemented
14#1543C211yesmaximal, characteristic, complemented
15#1540C211yesmaximal, characteristic, complemented
16#559—21no—
17#559C412yes—
18#550—21no—
19#550C412yes—
20#546—21no—
21#546—21no—
22#546C412yes—
23#546C412yes—
24#540—21no—
25#540—21no—
26#535—21no—
27#533C2211yescharacteristic
28#532C2211yescharacteristic
29#532C2211yescharacteristic
30#532C2211yescharacteristic
31#532C2211yescharacteristic
32C2×#127C2211yescharacteristic
33#518—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, complemented
44#432C2211yescharacteristic, complemented
45#431C2211yescharacteristic
46#431C2211yescharacteristic, complemented
47#431C2211yescharacteristic, complemented
48#423C2211yescharacteristic
49#423C2211yescharacteristic, complemented
50#423C2211yescharacteristic, complemented
51C2×C4⋊C8C2211yescharacteristic, complemented
52C2×C4⋊C8C2211yescharacteristic, complemented
53#408—21no—
54#405—21no—
55#333C2211yescharacteristic, complemented
56#333C2211yescharacteristic, complemented
57#333C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63#331C2211yescharacteristic
64#331C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67#331C2211yescharacteristic
68#331C2211yescharacteristic
69#331C2211yescharacteristic
70#331C2211yescharacteristic
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2×C412yes—
74C2×#41C2×C412yes—
75C2×#41D411yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4C2×C412yes—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—22no—
87#127—21no—
88#127—21no—
89#127C2×C412yes—
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#125—21no—
93#124—22no—
94#124—21no—
95#124—21no—
96#124C2×C412yes—
97#124C2×C412yes—
98#123—22no—
99#123—22no—
100#123—22no—
101#123—22no—
102C22⋊Q8—21no—
103C22⋊Q8C2×C412yes—
104#121—21no—
105#121—21no—
106#121—21no—
107#121—21no—
108#121—21no—
109#121—21no—
110#121—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#121C2×C412yes—
115#121C2×C412yes—
116#120—22no—
117#120—22no—
118C4×Q8—21no—
119C4×Q8C2×C412yes—
120C4×Q8C2×C411yescharacteristic
121C4×Q8D411yescharacteristic
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4C2×C412yes—
131C4×D4C2×C412yes—
132C4×D4C2×C411yescharacteristic
133C4×D4C2×C411yescharacteristic
134C4×D4C2×C411yescharacteristic
135C4×D4D411yescharacteristic
136C4×D4D411yescharacteristic
137C4×D4D411yescharacteristic
138#117—21no—
139#117D411yescharacteristic
140#117D411yescharacteristic
141#117D411yescharacteristic
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
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—
174C24—22no—
175#41—41no—
176#41—41no—
177#41—81no—
178#41—81no—
179#41—81no—
180#41—81no—
181#41—82no—
182#41—82no—
183C2×Q8—21no—
184C2×Q8C22⋊C411yescharacteristic
185C2×Q8C22×C412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—82no—
202C2×D4C22⋊C411yescharacteristic
203C2×D4C22⋊C411yescharacteristic
204C2×D4C22⋊C411yescharacteristic
205C2×D4C22×C412yes—
206C2×D4C22×C412yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4C22×C411yescharacteristic
212C22×C4C22×C411yescharacteristic
213C22×C4C22×C411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220M16—42no—
221M16—81no—
222M16—81no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—42no—
232C2×C8—81no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4C22⋊C411yescharacteristic
239C4⋊C4C22⋊C411yescharacteristic
240C4⋊C4C22⋊C411yescharacteristic
241C4⋊C4C22⋊C411yescharacteristic
242C4⋊C4C22⋊C411yescharacteristic
243C4⋊C4C22⋊C411yescharacteristic
244C4⋊C4C22×C412yes—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4C22⋊C411yescharacteristic
255C22⋊C4C22⋊C411yescharacteristic
256C22⋊C4C22⋊C411yescharacteristic
257C22⋊C4C22⋊C411yescharacteristic
258C22⋊C4C22⋊C411yescharacteristic
259C22⋊C4C22⋊C411yescharacteristic
260C42—22no—
261C42C22×C411yescharacteristic
262C42C22×C411yescharacteristic
263C42C22×C411yescharacteristic
264C42C2×D411yescharacteristic
265C42C2×D411yescharacteristic
266C42C2×D411yescharacteristic
267C42C2×D411yescharacteristic
268C42C2×D411yescharacteristic
269C42C2×D411yescharacteristic
270C42C2411yescharacteristic
271C23—22no—
272C23—22no—
273C23—44no—
274C23—82no—
275C23—82no—
276C23C2×C22⋊C411yescharacteristic
277C23C2×C22⋊C411yescharacteristic
278C23C2×C22⋊C411yescharacteristic
279Q8—41no—
280Q8—42no—
281Q8—41no—
282Q8—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—42no—
287D4—42no—
288D4—44no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—82no—
298D4—82no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×C22⋊C411yescharacteristic
325C2×C4C2×C22⋊C411yescharacteristic
326C2×C4C2×C22⋊C411yescharacteristic
327C2×C4C2×C22⋊C411yescharacteristic
328C2×C4C2×C22⋊C411yescharacteristic
329C2×C4C2×C22⋊C411yescharacteristic
330C2×C4C2×C22⋊C411yescharacteristic
331C2×C4C2×C22⋊C411yescharacteristic
332C2×C4C2×C22⋊C411yescharacteristic
333C2×C4C23×C411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C22×D411yescharacteristic
336C8—41no—
337C8—41no—
338C8—41no—
339C8—41no—
340C8—41no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—164no—
360C22C22×C22⋊C411yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—42no—
373C4#57511yescharacteristic
374C4#57511yescharacteristic
375C4#57611yescharacteristic
376C4#57711yescharacteristic
377C2—41no—
378C2—41no—
379C2—41no—
380C2—82no—
381C2—82no—
382C2#288811yescharacteristic
383C2#289611yescharacteristic
384C2#289711yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3508C2
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. #1560C2semidirect
5. #1559C2semidirect
6. #1557C2semidirect
7. #1555C2semidirect
8. #1554C2semidirect
9. #1553C2semidirect
10. #1551C2semidirect
11. #1551C2semidirect
12. #1547C2semidirect
13. #1545C2semidirect
14. #1543C2semidirect
15. #1540C2semidirect
43. #432C22semidirect
44. #432C22semidirect
46. #431C22semidirect
47. #431C22semidirect
49. #423C22semidirect
50. #423C22semidirect
51. C2×C4⋊C8C22semidirect
52. C2×C4⋊C8C22semidirect
55. #333C22semidirect
56. #333C22semidirect
57. #333C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
61. #332C22semidirect
62. #332C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#29892
#29252
#15602
#15592
#15572
#15552
#15542
#15532
#15512
#15472
#15452
#15432
#15402