Finite Group Databasebeta

← #17679

Subgroups — #17679

Order 256.

471 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514C211yesmaximal, characteristic
2#2970C211yesmaximal, characteristic, complemented
3#2923C211yesmaximal, characteristic, complemented
4#1536C211yesmaximal, characteristic, complemented
5#1535C211yesmaximal, characteristic, complemented
6#1535C211yesmaximal, characteristic, complemented
7#1533C211yesmaximal, characteristic, complemented
8#1532C211yesmaximal, characteristic, complemented
9#1532C211yesmaximal, characteristic, complemented
10#1530C211yesmaximal, characteristic, complemented
11#1529C211yesmaximal, characteristic, complemented
12#1527C211yesmaximal, characteristic, complemented
13#1527C211yesmaximal, characteristic, complemented
14#1524C211yesmaximal, characteristic, complemented
15C2×#329C211yesmaximal, characteristic, complemented
16#559C411yescharacteristic
17#556C411yescharacteristic
18#555—21no—
19#551—21no—
20#551—21no—
21D4×Q8C411yescharacteristic
22#548C411yescharacteristic
23#548C411yescharacteristic
24#546—21no—
25#546C411yescharacteristic
26#546C411yescharacteristic
27D42C411yescharacteristic
28#542—21no—
29#538—21no—
30#538—21no—
31#532C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#529C2211yescharacteristic
35C2×#124C2211yescharacteristic
36C2×#124C2211yescharacteristic
37#518—21no—
38C4×#41C2211yescharacteristic
39#446—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic, complemented
46#432C2211yescharacteristic, complemented
47#432C2211yescharacteristic, complemented
48#432C2211yescharacteristic, complemented
49#432C2211yescharacteristic, complemented
50#432C2211yescharacteristic, complemented
51#432C2211yescharacteristic, complemented
52#424—21no—
53#408—21no—
54C4×M16C2211yescharacteristic, complemented
55C4×M16C2211yescharacteristic, complemented
56C2×#97C2211yescharacteristic, complemented
57C2×#97C2211yescharacteristic, complemented
58C2×#97C2211yescharacteristic, complemented
59#330C2211yescharacteristic, complemented
60#330C2211yescharacteristic, complemented
61#330C2211yescharacteristic, complemented
62#330C2211yescharacteristic, complemented
63#329C2211yescharacteristic
64#329C2211yescharacteristic
65#329C2211yescharacteristic
66#329C2211yescharacteristic
67#329C2211yescharacteristic, complemented
68#329C2211yescharacteristic, complemented
69#329C2211yescharacteristic, complemented
70#329C2211yescharacteristic, complemented
71#329C2211yescharacteristic, complemented
72#329C2211yescharacteristic, complemented
73#329C2211yescharacteristic, complemented
74#329C2211yescharacteristic, complemented
75C2×#41C2×C411yescharacteristic
76C2×#41C2×C411yescharacteristic
77C2×#41D411yescharacteristic
78C22×Q8C2×C411yescharacteristic
79C22×Q8C2×C411yescharacteristic
80C22×D4—21no—
81C22×D4C2×C411yescharacteristic
82C22×D4C2×C411yescharacteristic
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C4⋊Q8—21no—
89C4⋊Q8C2×C411yescharacteristic
90#127C2×C411yescharacteristic
91#126—21no—
92#126—21no—
93#126—21no—
94#126—21no—
95#126—21no—
96#125C2311yescharacteristic
97#124—21no—
98#124—21no—
99#124—21no—
100#124—21no—
101#124—21no—
102#124—21no—
103#124—21no—
104#124C2×C411yescharacteristic
105#124C2×C411yescharacteristic
106#124C2×C411yescharacteristic
107#124C2×C411yescharacteristic
108#124C2311yescharacteristic
109#124C2311yescharacteristic
110#124C2311yescharacteristic
111#123—21no—
112#123—21no—
113#123—21no—
114#123—21no—
115#123—21no—
116#123—21no—
117#123—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8C2×C411yescharacteristic
124C22⋊Q8C2×C411yescharacteristic
125C22⋊Q8C2×C411yescharacteristic
126#121—21no—
127#121—21no—
128#121—21no—
129#121—21no—
130#121—21no—
131#121—21no—
132#121—21no—
133#121C2×C411yescharacteristic
134#121C2×C411yescharacteristic
135#121C2×C411yescharacteristic
136#120—21no—
137#120—21no—
138#120—21no—
139#120—21no—
140C4×Q8C2×C411yescharacteristic
141C4×Q8C2×C411yescharacteristic
142C4×Q8C2×C411yescharacteristic
143C4×Q8C2×C411yescharacteristic
144C4×Q8D411yescharacteristic
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4C2×C411yescharacteristic
155C4×D4C2×C411yescharacteristic
156C4×D4C2×C411yescharacteristic
157C4×D4C2×C411yescharacteristic
158C4×D4C2×C411yescharacteristic
159C4×D4C2×C411yescharacteristic
160C4×D4D411yescharacteristic
161C4×D4D411yescharacteristic
162C4×D4D411yescharacteristic
163#117—21no—
164#117—21no—
165#117—21no—
166#117—21no—
167#117D411yescharacteristic
168#117D411yescharacteristic
169#117D411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193#97—21no—
194#97—21no—
195#97—21no—
196#97C2311yescharacteristic
197#97C2311yescharacteristic
198#97C2311yescharacteristic
199#97C2311yescharacteristic
200#97C2311yescharacteristic
201#97C2311yescharacteristic
202#97C2311yescharacteristic
203#97C2311yescharacteristic
204C4×C8—21no—
205C4×C8—21no—
206C24—21no—
207C24—21no—
208#41—42no—
209#41—81no—
210#41—81no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×Q8C22⋊C411yescharacteristic
216C2×Q8C22×C411yescharacteristic
217C2×Q8C22×C411yescharacteristic
218C2×Q8C22×C411yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—44no—
229C2×D4—81no—
230C2×D4—81no—
231C2×D4C22⋊C411yescharacteristic
232C2×D4C22⋊C411yescharacteristic
233C2×D4C22⋊C411yescharacteristic
234C2×D4C22×C411yescharacteristic
235C2×D4C22×C411yescharacteristic
236C2×D4C22×C411yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4C22×C411yescharacteristic
245C22×C4C22×C411yescharacteristic
246C22×C4C22×C411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253M16—42no—
254M16—42no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4C22⋊C411yescharacteristic
277C4⋊C4C22⋊C411yescharacteristic
278C4⋊C4C22⋊C411yescharacteristic
279C4⋊C4C22⋊C411yescharacteristic
280C4⋊C4C22⋊C411yescharacteristic
281C4⋊C4C22⋊C411yescharacteristic
282C4⋊C4C22×C411yescharacteristic
283C4⋊C4C22×C411yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4C22⋊C411yescharacteristic
308C22⋊C4C22⋊C411yescharacteristic
309C22⋊C4C22⋊C411yescharacteristic
310C22⋊C4C22⋊C411yescharacteristic
311C22⋊C4C22⋊C411yescharacteristic
312C22⋊C4C22⋊C411yescharacteristic
313C42—21no—
314C42—21no—
315C42—21no—
316C42C22×C411yescharacteristic
317C42C22×C411yescharacteristic
318C42C22×C411yescharacteristic
319C42C2×D411yescharacteristic
320C42C2×D411yescharacteristic
321C42C2×D411yescharacteristic
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2×D411yescharacteristic
325C42C2411yescharacteristic
326C23—21no—
327C23—21no—
328C23—21no—
329C23—42no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C2×C22⋊C411yescharacteristic
336C23C2×C22⋊C411yescharacteristic
337C23C2×C22⋊C411yescharacteristic
338Q8—41no—
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—81no—
343Q8—81no—
344D4—22no—
345D4—22no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4C2×C22⋊C411yescharacteristic
381C2×C4C2×C22⋊C411yescharacteristic
382C2×C4C2×C22⋊C411yescharacteristic
383C2×C4C2×C22⋊C411yescharacteristic
384C2×C4C2×C22⋊C411yescharacteristic
385C2×C4C2×C22⋊C411yescharacteristic
386C2×C4C2×C22⋊C411yescharacteristic
387C2×C4C2×C22⋊C411yescharacteristic
388C2×C4C2×C22⋊C411yescharacteristic
389C2×C4C23×C411yescharacteristic
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C8—22no—
393C8—41no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—41no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—162no—
418C22—162no—
419C22#42112yes—
420C22C22×C22⋊C411yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C2—22no—
439C2—41no—
440C2—41no—
441C2—81no—
442C2—81no—
443C2—81no—
444C2#288811yescharacteristic
445C2C2×#42111yescharacteristic
446C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3514C2
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. #2970C2semidirect
3. #2923C2semidirect
4. #1536C2semidirect
5. #1535C2semidirect
6. #1535C2semidirect
7. #1533C2semidirect
8. #1532C2semidirect
9. #1532C2semidirect
10. #1530C2semidirect
11. #1529C2semidirect
12. #1527C2semidirect
13. #1527C2semidirect
14. #1524C2semidirect
15. C2×#329C2semidirect
45. #432C22semidirect
46. #432C22semidirect
47. #432C22semidirect
48. #432C22semidirect
49. #432C22semidirect
50. #432C22semidirect
51. #432C22semidirect
54. C4×M16C22semidirect
55. C4×M16C22semidirect
56. C2×#97C22semidirect
57. C2×#97C22semidirect
58. C2×#97C22semidirect
59. #330C22semidirect
60. #330C22semidirect
61. #330C22semidirect
62. #330C22semidirect
67. #329C22semidirect
68. #329C22semidirect
69. #329C22semidirect
70. #329C22semidirect
71. #329C22semidirect
72. #329C22semidirect
73. #329C22semidirect
74. #329C22semidirect

Maximal subgroups

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

subgroupindex
#35142
#29702
#29232
#15362
#15352
#15332
#15322
#15302
#15292
#15272
#15242
C2×#3292