Finite Group Databasebeta

← #17516

Subgroups — #17516

Order 256.

474 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3480C211yesmaximal, characteristic
2#2965C211yesmaximal, characteristic, complemented
3#2925C211yesmaximal, characteristic, complemented
4#1496C211yesmaximal, characteristic, complemented
5#1495C211yesmaximal, characteristic, complemented
6#1494C211yesmaximal, characteristic, complemented
7#1491C211yesmaximal, characteristic, complemented
8#1489C211yesmaximal, characteristic, complemented
9#1488C211yesmaximal, characteristic, complemented
10#1484C211yesmaximal, characteristic, complemented
11#1483C211yesmaximal, characteristic, complemented
12#1482C211yesmaximal, characteristic, complemented
13#1479C211yesmaximal, characteristic, complemented
14#1477C211yesmaximal, characteristic, complemented
15#1476C211yesmaximal, characteristic, complemented
16#559—21no—
17#559C411yescharacteristic
18#557C411yescharacteristic
19#556—21no—
20#556C411yescharacteristic
21#553—21no—
22#553—21no—
23#553C411yescharacteristic
24#553C411yescharacteristic
25#550C411yescharacteristic
26#548C411yescharacteristic
27#548C411yescharacteristic
28#532—21no—
29#532C2211yescharacteristic
30#529—21no—
31#529—21no—
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34C4×#41—21no—
35C4×#41C2211yescharacteristic
36C4×#41C2211yescharacteristic
37C4×#41C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39#446—21no—
40#436—21no—
41#435—21no—
42#435—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic
46#432C2211yescharacteristic
47#431C2211yescharacteristic
48#431C2211yescharacteristic
49#431C2211yescharacteristic
50#431C2211yescharacteristic
51#431C2211yescharacteristic
52#423C2211yescharacteristic
53C2×C4⋊C8C2211yescharacteristic
54#408—21no—
55#405—21no—
56C4×M16C2211yescharacteristic
57C4×M16C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59Q8⋊C8C2211yescharacteristic
60Q8⋊C8C2211yescharacteristic
61Q8⋊C8C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65Q8⋊C8C2211yescharacteristic
66Q8⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71D4⋊C8C2211yescharacteristic
72D4⋊C8C2211yescharacteristic
73D4⋊C8C2211yescharacteristic
74D4⋊C8C2211yescharacteristic
75C2×#41—21no—
76C2×#41C2×C411yescharacteristic
77C2×#41C2×C411yescharacteristic
78C2×#41D411yescharacteristic
79C22×Q8C2×C411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8—21no—
86C4⋊Q8C2×C411yescharacteristic
87#127—21no—
88#127C2×C411yescharacteristic
89#126—21no—
90#126—21no—
91#126—21no—
92#126—21no—
93#126—21no—
94#126—21no—
95#125—21no—
96#125—21no—
97#125C2×C411yescharacteristic
98#125C2×C411yescharacteristic
99#124—21no—
100#124—21no—
101#124—21no—
102#124—21no—
103#124—21no—
104#124C2×C411yescharacteristic
105#124C2×C411yescharacteristic
106#123—21no—
107#123—21no—
108#123—21no—
109#123—21no—
110#123—21no—
111#123—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8C2×C411yescharacteristic
115C22⋊Q8C2×C411yescharacteristic
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121C2×C411yescharacteristic
124#121C2×C411yescharacteristic
125#121C2×C411yescharacteristic
126#121C2×C411yescharacteristic
127C4×Q8—21no—
128C4×Q8—21no—
129C4×Q8—21no—
130C4×Q8—21no—
131C4×Q8C2×C411yescharacteristic
132C4×Q8C2×C411yescharacteristic
133C4×Q8C2×C411yescharacteristic
134C4×Q8C2×C411yescharacteristic
135C4×Q8C2×C411yescharacteristic
136C4×Q8D411yescharacteristic
137C4×Q8C2311yescharacteristic
138C4×Q8C2311yescharacteristic
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×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4C2×C411yescharacteristic
153C4×D4C2×C411yescharacteristic
154C4×D4C2×C411yescharacteristic
155C4×D4C2×C411yescharacteristic
156C4×D4C2×C411yescharacteristic
157C4×D4D411yescharacteristic
158C4×D4D411yescharacteristic
159C4×D4D411yescharacteristic
160C4×D4C2311yescharacteristic
161C4×D4C2311yescharacteristic
162#117—21no—
163#117—21no—
164#117—21no—
165#117—21no—
166#117C2×C411yescharacteristic
167#117C2×C411yescharacteristic
168#117D411yescharacteristic
169#117D411yescharacteristic
170#117D411yescharacteristic
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C411yescharacteristic
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8C2311yescharacteristic
185C4⋊C8C2311yescharacteristic
186C4⋊C8C2311yescharacteristic
187C4⋊C8C2311yescharacteristic
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195#97—21no—
196#97—21no—
197#97—21no—
198#97—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C2311yescharacteristic
203C4×C8C2311yescharacteristic
204C4×C8C2311yescharacteristic
205C4×C8C2311yescharacteristic
206#41—22no—
207#41—22no—
208#41—41no—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—42no—
213#41—42no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8C22⋊C411yescharacteristic
219C2×Q8C22×C411yescharacteristic
220C2×Q8C22×C411yescharacteristic
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4C22⋊C411yescharacteristic
230C2×D4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
243C22×C4C22×C411yescharacteristic
244C22×C4C22×C411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251M16—42no—
252M16—42no—
253M16—42no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—21no—
265C4⋊C4—21no—
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—21no—
278C4⋊C4C22⋊C411yescharacteristic
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
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⋊C4C22⋊C411yescharacteristic
305C22⋊C4C22⋊C411yescharacteristic
306C22⋊C4C22⋊C411yescharacteristic
307C22⋊C4C22⋊C411yescharacteristic
308C22⋊C4C22⋊C411yescharacteristic
309C22⋊C4C22⋊C411yescharacteristic
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42C22×C411yescharacteristic
319C42C22×C411yescharacteristic
320C42C22×C411yescharacteristic
321C42C2×D411yescharacteristic
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2×D411yescharacteristic
325C42C2×D411yescharacteristic
326C42C2×D411yescharacteristic
327C42C2411yescharacteristic
328C23—21no—
329C23—21no—
330C23C2×C22⋊C411yescharacteristic
331C23C2×C22⋊C411yescharacteristic
332C23C2×C22⋊C411yescharacteristic
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8—41no—
337Q8—42no—
338Q8—42no—
339Q8#13112yes—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354D4—81no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×C22⋊C411yescharacteristic
387C2×C4C2×C22⋊C411yescharacteristic
388C2×C4C2×C22⋊C411yescharacteristic
389C2×C4C2×C22⋊C411yescharacteristic
390C2×C4C2×C22⋊C411yescharacteristic
391C2×C4C2×C22⋊C411yescharacteristic
392C2×C4C2×C22⋊C411yescharacteristic
393C2×C4C2×C22⋊C411yescharacteristic
394C2×C4C2×C22⋊C411yescharacteristic
395C2×C4C23×C411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C8—22no—
399C8—41no—
400C8—41no—
401C8—41no—
402C8—41no—
403C8—41no—
404C8—41no—
405C8—41no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22C22×C22⋊C411yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4C2×#13111yescharacteristic
439C4#56811yescharacteristic
440C4#57511yescharacteristic
441C4#57511yescharacteristic
442C2—41no—
443C2—41no—
444C2—41no—
445C2—81no—
446C2—81no—
447C2#288011yescharacteristic
448C2#289611yescharacteristic
449C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2965C2semidirect
3. #2925C2semidirect
4. #1496C2semidirect
5. #1495C2semidirect
6. #1494C2semidirect
7. #1491C2semidirect
8. #1489C2semidirect
9. #1488C2semidirect
10. #1484C2semidirect
11. #1483C2semidirect
12. #1482C2semidirect
13. #1479C2semidirect
14. #1477C2semidirect
15. #1476C2semidirect

Maximal subgroups

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

subgroupindex
#34802
#29652
#29252
#14962
#14952
#14942
#14912
#14892
#14882
#14842
#14832
#14822
#14792
#14772
#14762