Finite Group Databasebeta

← #17604

Subgroups — #17604

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3494C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic, complemented
3#2343C211yesmaximal, characteristic, complemented
4#1521C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1516C211yesmaximal, characteristic, complemented
7#1514C211yesmaximal, characteristic, complemented
8#1512C211yesmaximal, characteristic, complemented
9#1510C211yesmaximal, characteristic, complemented
10#1509C211yesmaximal, characteristic, complemented
11#1505C211yesmaximal, characteristic, complemented
12#1504C211yesmaximal, characteristic, complemented
13#1502C211yesmaximal, characteristic, complemented
14#1500C211yesmaximal, characteristic, complemented
15C2×#327C211yesmaximal, characteristic, complemented
16#555—21no—
17#553—21no—
18#553—21no—
19#551—21no—
20#551—21no—
21#547—21no—
22#546—21no—
23#546—21no—
24#544C411yescharacteristic, complemented
25#542C411yescharacteristic, complemented
26#542C411yescharacteristic, complemented
27#540C411yescharacteristic, complemented
28#540C411yescharacteristic, complemented
29#535C411yescharacteristic, complemented
30#532C2211yescharacteristic
31#532C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#525C2211yescharacteristic
35C2×#121C2211yescharacteristic
36#520C411yescharacteristic, complemented
37#518C411yescharacteristic, complemented
38C2×#117C2211yescharacteristic
39#433—21no—
40#433—21no—
41#432C2211yescharacteristic, complemented
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44#431C2211yescharacteristic, complemented
45#424—21no—
46#407C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#405—21no—
49#387—21no—
50#387—21no—
51#385—21no—
52#383C2211yescharacteristic, complemented
53#382C2211yescharacteristic, complemented
54#381C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic, complemented
56C4×C22⋊C4—21no—
57#376C2211yescharacteristic, complemented
58C2×#95C2211yescharacteristic
59#328C2211yescharacteristic, complemented
60#328C2211yescharacteristic, complemented
61#328C2211yescharacteristic, complemented
62#328C2211yescharacteristic, complemented
63#327C2211yescharacteristic
64#327C2211yescharacteristic
65#327C2211yescharacteristic
66#327C2211yescharacteristic
67#327C2211yescharacteristic
68#327C2211yescharacteristic
69#327C2211yescharacteristic
70#327C2211yescharacteristic
71#327C2211yescharacteristic, complemented
72#327C2211yescharacteristic, complemented
73#327C2211yescharacteristic, complemented
74#327C2211yescharacteristic, complemented
75C2×#41—21no—
76C2×#41—21no—
77C2×#41C2×C411yescharacteristic
78C22×D4—21no—
79C22×D4C2×C411yescharacteristic
80C23×C4—21no—
81C23×C4C2311yescharacteristic
82C2×M16—21no—
83C22×C8—21no—
84C4⋊Q8—21no—
85C4⋊Q8C2×C411yescharacteristic, complemented
86C4⋊Q8C2×C411yescharacteristic, complemented
87#127C2×C411yescharacteristic, complemented
88#127C2×C411yescharacteristic, complemented
89#126—21no—
90#126—21no—
91#126—21no—
92#126—21no—
93#125—21no—
94#125C2×C411yescharacteristic, complemented
95#125C2×C411yescharacteristic, complemented
96#124—21no—
97#124—21no—
98#124—21no—
99#124—21no—
100#124C2×C411yescharacteristic, complemented
101#124C2×C411yescharacteristic, complemented
102#123—21no—
103#123—21no—
104#123—21no—
105#123—21no—
106#123—21no—
107#123—21no—
108#123—21no—
109#123—21no—
110#123—21no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C22⋊Q8C2×C411yescharacteristic
116C22⋊Q8C2311yescharacteristic
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121C2×C411yescharacteristic
128#121C2×C411yescharacteristic
129#121C2×C411yescharacteristic
130#121C2311yescharacteristic
131#121C2311yescharacteristic
132#121C2311yescharacteristic
133#120—21no—
134#120—21no—
135#120—21no—
136C4×Q8C2×C411yescharacteristic, complemented
137C4×Q8C2×C411yescharacteristic, complemented
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4C2×C411yescharacteristic, complemented
149C4×D4C2×C411yescharacteristic, complemented
150C4×D4C2×C411yescharacteristic, complemented
151C4×D4C2×C411yescharacteristic, complemented
152C4×D4C2×C411yescharacteristic, complemented
153C4×D4C2×C411yescharacteristic, complemented
154#117—21no—
155#117—21no—
156#117—21no—
157#117C2×C411yescharacteristic
158#117C2×C411yescharacteristic
159#117C2×C411yescharacteristic
160#117C2×C411yescharacteristic
161#117C2×C411yescharacteristic
162#117D411yescharacteristic
163#117D411yescharacteristic
164#117D411yescharacteristic
165#117D411yescharacteristic
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×C411yescharacteristic
170C2×C4⋊C4D411yescharacteristic
171C2×C4⋊C4D411yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4D411yescharacteristic
177C2×C22⋊C4D411yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8C2311yescharacteristic
189C22⋊C8C2311yescharacteristic
190C22⋊C8C2311yescharacteristic
191C22⋊C8C2311yescharacteristic
192#97—21no—
193#97—21no—
194C4×C8—21no—
195C4×C8—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2311yescharacteristic
203#95C2311yescharacteristic
204#95C2311yescharacteristic
205#95C2311yescharacteristic
206C24—21no—
207C24C2×D411yescharacteristic
208#41—42no—
209#41—81no—
210#41—81no—
211#41—81no—
212#41—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8C22×C411yescharacteristic
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
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—42no—
229C2×D4—81no—
230C2×D4—81no—
231C2×D4C22×C411yescharacteristic
232C2×D4C22×C411yescharacteristic
233C2×D4C22×C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4C22×C411yescharacteristic
251C22×C4C22×C411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2411yescharacteristic
260M16—42no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
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⋊C4—21no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4C22⋊C411yescharacteristic
280C4⋊C4C22⋊C411yescharacteristic
281C4⋊C4C22⋊C411yescharacteristic
282C4⋊C4C22⋊C411yescharacteristic
283C4⋊C4C22⋊C411yescharacteristic
284C4⋊C4C22⋊C411yescharacteristic
285C4⋊C4C22⋊C411yescharacteristic
286C4⋊C4C22⋊C411yescharacteristic
287C4⋊C4C22×C411yescharacteristic
288C4⋊C4C22×C411yescharacteristic
289C4⋊C4C22×C411yescharacteristic
290C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—42no—
311C22⋊C4C22⋊C411yescharacteristic
312C22⋊C4C22⋊C411yescharacteristic
313C22⋊C4C22⋊C411yescharacteristic
314C22⋊C4C22⋊C411yescharacteristic
315C22⋊C4C22⋊C411yescharacteristic
316C22⋊C4C22⋊C411yescharacteristic
317C22⋊C4C22⋊C411yescharacteristic
318C22⋊C4C22⋊C411yescharacteristic
319C42—21no—
320C42—21no—
321C42—21no—
322C42—42no—
323C42—81no—
324C42C22×C411yescharacteristic
325C42C22×C411yescharacteristic
326C42C22×C411yescharacteristic
327C42C22×C411yescharacteristic
328C42C2×D411yescharacteristic
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2×D411yescharacteristic
332C23—21no—
333C23—21no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—41no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23C2×C22⋊C411yescharacteristic
343C23C2×C22⋊C411yescharacteristic
344C23C22×D411yescharacteristic
345Q8—41no—
346Q8—81no—
347Q8—81no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—84no—
404C2×C4C2×C22⋊C411yescharacteristic
405C2×C4C2×C22⋊C411yescharacteristic
406C2×C4C2×C22⋊C411yescharacteristic
407C2×C4C2×C22⋊C411yescharacteristic
408C2×C4C2×C22⋊C411yescharacteristic
409C2×C4C2×C22⋊C411yescharacteristic
410C2×C4C2×C22⋊C411yescharacteristic
411C2×C4C2×C22⋊C411yescharacteristic
412C2×C4C2×C22⋊C411yescharacteristic
413C2×C4C2×C22⋊C411yescharacteristic
414C2×C4C23×C411yescharacteristic
415C2×C4C22×D411yescharacteristic
416C8—41no—
417C8—41no—
418C8—41no—
419C8—41no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—162no—
439C22#41012yes—
440C22C22×C22⋊C411yescharacteristic
441C22#57611yescharacteristic
442C22#57711yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—81no—
462C4—81no—
463C4—81no—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—41no—
468C2—81no—
469C2—81no—
470C2—81no—
471C2C2×#41011yescharacteristic
472C2#289711yescharacteristic
473C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3494C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2924C2semidirect
3. #2343C2semidirect
4. #1521C2semidirect
5. #1517C2semidirect
6. #1516C2semidirect
7. #1514C2semidirect
8. #1512C2semidirect
9. #1510C2semidirect
10. #1509C2semidirect
11. #1505C2semidirect
12. #1504C2semidirect
13. #1502C2semidirect
14. #1500C2semidirect
15. C2×#327C2semidirect
24. #544C4semidirect
25. #542C4semidirect
26. #542C4semidirect
27. #540C4semidirect
28. #540C4semidirect
29. #535C4semidirect
36. #520C4semidirect
37. #518C4semidirect
41. #432C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. #431C22semidirect
52. #383C22semidirect
53. #382C22semidirect
55. C4×C4⋊C4C22semidirect
57. #376C22semidirect
59. #328C22semidirect
60. #328C22semidirect
61. #328C22semidirect
62. #328C22semidirect
71. #327C22semidirect
72. #327C22semidirect
73. #327C22semidirect
74. #327C22semidirect
85. C4⋊Q8C2×C4semidirect
86. C4⋊Q8C2×C4semidirect
87. #127C2×C4semidirect
88. #127C2×C4semidirect
94. #125C2×C4semidirect
95. #125C2×C4semidirect
100. #124C2×C4semidirect
101. #124C2×C4semidirect
136. C4×Q8C2×C4semidirect
137. C4×Q8C2×C4semidirect
148. C4×D4C2×C4semidirect
149. C4×D4C2×C4semidirect
150. C4×D4C2×C4semidirect
151. C4×D4C2×C4semidirect
152. C4×D4C2×C4semidirect
153. C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34942
#29242
#23432
#15212
#15172
#15162
#15142
#15122
#15102
#15092
#15052
#15042
#15022
#15002
C2×#3272