Finite Group Databasebeta

← #17524

Subgroups — #17524

Order 256.

468 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3482C211yesmaximal, characteristic
2#2990C211yesmaximal, characteristic, complemented
3#2922C211yesmaximal, characteristic, complemented
4#1497C211yesmaximal, characteristic, complemented
5#1495C211yesmaximal, characteristic, complemented
6#1494C211yesmaximal, characteristic, complemented
7#1491C211yesmaximal, characteristic, complemented
8#1488C211yesmaximal, characteristic, complemented
9#1487C211yesmaximal, characteristic, complemented
10#1483C211yesmaximal, characteristic, complemented
11#1482C211yesmaximal, characteristic, complemented
12#1481C211yesmaximal, characteristic, complemented
13#1479C211yesmaximal, characteristic, complemented
14#1477C211yesmaximal, characteristic, complemented
15C2×Q8⋊C8C211yesmaximal, characteristic, complemented
16#557C411yescharacteristic
17#556—21no—
18#556C411yescharacteristic
19#552—21no—
20#552—21no—
21#552C411yescharacteristic
22#552C411yescharacteristic
23D4×Q8C411yescharacteristic
24#548C411yescharacteristic
25#548C411yescharacteristic
26#547—21no—
27#547C411yescharacteristic
28#533—21no—
29#533C2211yescharacteristic
30#529—21no—
31#529—21no—
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34C4×#41—21no—
35C4×#41C2211yescharacteristic
36C4×#41C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39#446—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic
46#432C2211yescharacteristic
47#431C2211yescharacteristic
48#431C2211yescharacteristic
49#431C2211yescharacteristic
50#431C2211yescharacteristic
51#431C2211yescharacteristic
52#424—21no—
53#423C2211yescharacteristic
54C2×C4⋊C8C2211yescharacteristic
55C2×C4⋊C8C2211yescharacteristic
56#408—21no—
57C4×M16C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59Q8⋊C8C2211yescharacteristic
60Q8⋊C8C2211yescharacteristic
61Q8⋊C8C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65Q8⋊C8C2211yescharacteristic
66Q8⋊C8C2211yescharacteristic
67Q8⋊C8C2211yescharacteristic
68Q8⋊C8C2211yescharacteristic
69Q8⋊C8C2211yescharacteristic
70Q8⋊C8C2211yescharacteristic
71D4⋊C8C2211yescharacteristic
72D4⋊C8C2211yescharacteristic
73D4⋊C8C2211yescharacteristic
74D4⋊C8C2211yescharacteristic
75C2×#41—21no—
76C2×#41C2×C411yescharacteristic
77C2×#41D411yescharacteristic
78C22×Q8C2×C411yescharacteristic
79C22×Q8C2×C411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8C2×C411yescharacteristic
88C4⋊Q8C2×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#125—21no—
98#125—21no—
99#125—21no—
100#125C2×C411yescharacteristic
101#125C2×C411yescharacteristic
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⋊Q8—21no—
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8C2×C411yescharacteristic
120C22⋊Q8C2×C411yescharacteristic
121C22⋊Q8C2×C411yescharacteristic
122C22⋊Q8C2×C411yescharacteristic
123#121—21no—
124#121—21no—
125#121C2×C411yescharacteristic
126#121C2×C411yescharacteristic
127C4×Q8—21no—
128C4×Q8—21no—
129C4×Q8—21no—
130C4×Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8C2×C411yescharacteristic
134C4×Q8C2×C411yescharacteristic
135C4×Q8C2×C411yescharacteristic
136C4×Q8C2×C411yescharacteristic
137C4×Q8C2×C411yescharacteristic
138C4×Q8C2×C411yescharacteristic
139C4×Q8D411yescharacteristic
140C4×Q8C2311yescharacteristic
141C4×Q8C2311yescharacteristic
142C4×Q8C2311yescharacteristic
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4C2×C411yescharacteristic
151C4×D4C2×C411yescharacteristic
152C4×D4C2×C411yescharacteristic
153C4×D4C2×C411yescharacteristic
154C4×D4D411yescharacteristic
155C4×D4D411yescharacteristic
156C4×D4D411yescharacteristic
157C4×D4C2311yescharacteristic
158#117—21no—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117—21no—
164#117—21no—
165#117C2×C411yescharacteristic
166#117D411yescharacteristic
167#117D411yescharacteristic
168#117D411yescharacteristic
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C2×C411yescharacteristic
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⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8C2311yescharacteristic
187C4⋊C8C2311yescharacteristic
188C4⋊C8C2311yescharacteristic
189C4⋊C8C2311yescharacteristic
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197#97—21no—
198#97—21no—
199#97—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C2311yescharacteristic
203C4×C8C2311yescharacteristic
204C4×C8C2311yescharacteristic
205C4×C8C2311yescharacteristic
206#41—22no—
207#41—42no—
208#41—41no—
209#41—44no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8C22⋊C411yescharacteristic
218C2×Q8C22×C411yescharacteristic
219C2×Q8C22×C411yescharacteristic
220C2×Q8C22×C411yescharacteristic
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C22⋊C411yescharacteristic
225C2×D4C22⋊C411yescharacteristic
226C2×D4C22⋊C411yescharacteristic
227C2×D4C22×C411yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C22×C411yescharacteristic
236C22×C4C22×C411yescharacteristic
237C22×C4C22×C411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244M16—42no—
245M16—81no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—42no—
255C2×C8—81no—
256C2×C8—81no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
302C22⋊C4C22⋊C411yescharacteristic
303C22⋊C4C22⋊C411yescharacteristic
304C22⋊C4C22⋊C411yescharacteristic
305C22⋊C4C22⋊C411yescharacteristic
306C22⋊C4C22⋊C411yescharacteristic
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
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—
327C23C2×C22⋊C411yescharacteristic
328C23C2×C22⋊C411yescharacteristic
329C23C2×C22⋊C411yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—41no—
336Q8—42no—
337Q8—42no—
338Q8—42no—
339Q8#13112yes—
340D4—22no—
341D4—22no—
342D4—21no—
343D4—21no—
344D4—41no—
345D4—42no—
346D4—42no—
347D4—42no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
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—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4C2×C22⋊C411yescharacteristic
379C2×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C8—41no—
391C8—41no—
392C8—41no—
393C8—41no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22#42112yes—
410C22C22×C22⋊C411yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4C2×#13111yescharacteristic
429C4#56811yescharacteristic
430C4#57711yescharacteristic
431C4#57811yescharacteristic
432C2—22no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2#288011yescharacteristic
437C2#289811yescharacteristic
438C2C2×#42111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3482C2
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. #2990C2semidirect
3. #2922C2semidirect
4. #1497C2semidirect
5. #1495C2semidirect
6. #1494C2semidirect
7. #1491C2semidirect
8. #1488C2semidirect
9. #1487C2semidirect
10. #1483C2semidirect
11. #1482C2semidirect
12. #1481C2semidirect
13. #1479C2semidirect
14. #1477C2semidirect
15. C2×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#34822
#29902
#29222
#14972
#14952
#14942
#14912
#14882
#14872
#14832
#14822
#14812
#14792
#14772
C2×Q8⋊C82