Finite Group Databasebeta

← #60975

Subgroups — #60975

Order 256.

834 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3461C212yesmaximal, complemented
2#3263C211yesmaximal, characteristic, complemented
3#3261C211yesmaximal, characteristic, complemented
4#3258C212yesmaximal, complemented
5#3244C212yesmaximal, complemented
6#3242C212yesmaximal, complemented
7#3225C212yesmaximal, complemented
8#3223C212yesmaximal, complemented
9#3204C211yesmaximal, characteristic
10#3202C211yesmaximal, characteristic, complemented
11#3201C212yesmaximal, complemented
12#3185C211yesmaximal, characteristic, complemented
13#3183C211yesmaximal, characteristic, complemented
14#3181C212yesmaximal, complemented
15#3117C212yesmaximal, complemented
16#3116C212yesmaximal, complemented
17#3113C212yesmaximal, complemented
18#3111C212yesmaximal, complemented
19#2923C211yesmaximal, characteristic, complemented
20#544C2212yes—
21#542—22no—
22#542C2212yes—
23#542C2212yes—
24#541—22no—
25#541C2212yes—
26#541C2212yes—
27#539—22no—
28#539—22no—
29#537C2212yes—
30#536C2212yes—
31#536C2212yes—
32#533—22no—
33#533C2212yes—
34#533C2212yes—
35#529—22no—
36#529—22no—
37#529—22no—
38#529C2212yes—
39#529C2212yes—
40C4×#41C2212yes—
41C4×#41C2211yescharacteristic
42C2×C4×Q8C2212yes—
43#501C2212yes—
44#499C2211yescharacteristic
45#498C2211yescharacteristic
46#497C2212yes—
47#495C2211yescharacteristic
48#494C2211yescharacteristic
49#491C2212yes—
50#489C2212yes—
51#488C2212yes—
52#487C2212yes—
53#485C2212yes—
54#485C2212yes—
55#485C2212yes—
56#484C2212yes—
57#483C2212yes—
58#483C2212yes—
59#482C2212yes—
60#481C2212yes—
61#479C2212yes—
62#479C2212yes—
63#479C2212yes—
64#478C2212yes—
65#477C2212yes—
66#476C2212yes—
67#475C2212yes—
68#475C2212yes—
69#470C2211yescharacteristic
70#470C2212yes—
71#470C2211yescharacteristic
72#470C2211yescharacteristic
73#469C2211yescharacteristic
74#468C2212yes—
75#467C2212yes—
76#467C2212yes—
77#465C2212yes—
78#465C2212yes—
79#464C2212yes—
80#464C2212yes—
81#464C2212yes—
82#463C2212yes—
83#462C2212yes—
84#462C2212yes—
85#462C2212yes—
86#461C2212yes—
87#452C2212yes—
88#452C2212yes—
89#452C2212yes—
90#451C2212yes—
91#451C2212yes—
92#451C2212yes—
93#449C2212yes—
94#448C2212yes—
95#442C2211yescharacteristic
96#441C2212yes—
97#441C2211yescharacteristic
98#441C2212yes—
99#441C2212yes—
100#441C2211yescharacteristic
101#441C2211yescharacteristic
102#440C2212yes—
103#440C2212yes—
104#440C2212yes—
105#433C2212yes—
106#433C2212yes—
107#432C2212yes—
108#432C2212yes—
109#424C2211yescharacteristic
110#424C2211yescharacteristic
111#408C2211yescharacteristic
112#408C2211yescharacteristic
113C4×M16C2211yescharacteristic
114C2×#97C2211yescharacteristic
115C2×#41C2312yes—
116C2×#41C2311yescharacteristic
117C22×Q8C2312yes—
118C2×Q16C2311yescharacteristic
119C2×Q16C2312yes—
120C2×Q16C2311yescharacteristic
121C2×Q16C2312yes—
122C2×Q16C2312yes—
123C2×Q16C2311yescharacteristic
124C2×SD16C2312yes—
125C2×SD16C2312yes—
126C2×SD16C2312yes—
127C2×D8C2311yescharacteristic
128C2×M16C2311yescharacteristic
129C2×M16C2311yescharacteristic
130C22×C8C2311yescharacteristic
131C22×C8C2311yescharacteristic
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8C2312yes—
135C4⋊Q8C2312yes—
136C4⋊Q8C2312yes—
137C4⋊Q8C2312yes—
138#126—22no—
139#126—22no—
140#126—22no—
141#126—22no—
142#126—22no—
143#126—22no—
144#126—22no—
145#126—22no—
146#125—22no—
147#125—22no—
148#125—22no—
149#125—22no—
150#125—22no—
151#125—22no—
152#125—22no—
153#125C2312yes—
154#125C2312yes—
155#124—22no—
156#124—22no—
157#124—22no—
158#124C2312yes—
159#124C2312yes—
160#123—22no—
161#123—22no—
162#123—22no—
163#123—22no—
164#123—22no—
165#123—22no—
166#123—22no—
167#123—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8C2312yes—
178C22⋊Q8C2312yes—
179C22⋊Q8C2312yes—
180C22⋊Q8C2312yes—
181C22⋊Q8C2312yes—
182C22⋊Q8C2312yes—
183#121—22no—
184#121—22no—
185#121—22no—
186#121C2312yes—
187#121C2312yes—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8D412yes—
193C4×Q8C2312yes—
194C4×Q8C2312yes—
195C4×Q8C2312yes—
196C4×Q8C2312yes—
197C4×Q8C2312yes—
198C4×Q8C2312yes—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4D412yes—
204C4×D4D412yes—
205C4×D4D412yes—
206C4×D4C2312yes—
207C4×D4C2312yes—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117C2312yes—
214#117C2312yes—
215#117C2311yescharacteristic
216C2×C4⋊C4—22no—
217C2×C4⋊C4C2312yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2312yes—
221C2×C42C2311yescharacteristic
222#107C2312yes—
223#107C2312yes—
224#107C2312yes—
225#106C2312yes—
226#106C2312yes—
227#106C2311yescharacteristic
228#106C2312yes—
229#106C2311yescharacteristic
230#106C2311yescharacteristic
231#106C2311yescharacteristic
232C4⋊C8C2312yes—
233C4⋊C8C2312yes—
234C4⋊C8C2312yes—
235C4⋊C8C2312yes—
236#103C2312yes—
237#103C2312yes—
238#103C2312yes—
239#103C2312yes—
240#103C2312yes—
241#103C2312yes—
242#103C2312yes—
243#103C2312yes—
244#103C2312yes—
245#103C2312yes—
246#103C2312yes—
247#103C2312yes—
248#102C2312yes—
249#102C2312yes—
250#102C2312yes—
251#102C2312yes—
252C22⋊C8C2312yes—
253C22⋊C8C2312yes—
254C22⋊C8C2312yes—
255C22⋊C8C2312yes—
256#97C2312yes—
257#97C2312yes—
258#97C2311yescharacteristic
259#97C2311yescharacteristic
260C4×C8C2311yescharacteristic
261C4×C8C2311yescharacteristic
262#41—41no—
263#41—41no—
264#41—42no—
265#41—42no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8C2×D411yescharacteristic
272C2×Q8C2412yes—
273C2×Q8C2412yes—
274C2×Q8C2412yes—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4C2×D411yescharacteristic
279C2×D4C2×D412yes—
280C2×D4C2412yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4C2×D412yes—
287C22×C4C2×D412yes—
288C22×C4C2×D412yes—
289C22×C4C2412yes—
290C22×C4C2411yescharacteristic
291Q16—42no—
292Q16—42no—
293Q16—41no—
294Q16—41no—
295Q16—42no—
296Q16—41no—
297SD16—42no—
298SD16—42no—
299SD16—42no—
300D8—41no—
301M16—41no—
302M16—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8C2411yescharacteristic
306C2×C8C2411yescharacteristic
307C2×C8C2411yescharacteristic
308C2×C8C2411yescharacteristic
309C2×C8C2412yes—
310C2×C8C2412yes—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4C2×D412yes—
335C4⋊C4C2×D412yes—
336C4⋊C4C2×D412yes—
337C4⋊C4C2412yes—
338C4⋊C4C2412yes—
339C4⋊C4C2412yes—
340C4⋊C4C2412yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4C2×D412yes—
357C22⋊C4C2×D412yes—
358C22⋊C4C2×D411yescharacteristic
359C22⋊C4C2×D411yescharacteristic
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42C2×D412yes—
368C42C2×D412yes—
369C42C2×D412yes—
370C42C2411yescharacteristic
371C42C2411yescharacteristic
372C42C2412yes—
373C23—22no—
374C23C22×D411yescharacteristic
375C23C22×D412yes—
376Q8—22no—
377Q8—22no—
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8—22no—
382Q8—41no—
383Q8—44no—
384Q8—82no—
385Q8—82no—
386D4—22no—
387D4—22no—
388D4—42no—
389D4—41no—
390D4—44no—
391D4—82no—
392D4—82no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4C22×D412yes—
413C2×C4C22×D412yes—
414C2×C4C22×D412yes—
415C2×C4C22×D412yes—
416C2×C4C22×D411yescharacteristic
417C2×C4C22×D412yes—
418C2×C4C2511yescharacteristic
419C8—22no—
420C8—22no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22C23×D411yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4#58511yescharacteristic
448C4#58511yescharacteristic
449C2—42no—
450C2—41no—
451C2—82no—
452C2#344911yescharacteristic
453C2#358612yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3204C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3461C2semidirect
2. #3263C2semidirect
3. #3261C2semidirect
4. #3258C2semidirect
5. #3244C2semidirect
6. #3242C2semidirect
7. #3225C2semidirect
8. #3223C2semidirect
10. #3202C2semidirect
11. #3201C2semidirect
12. #3185C2semidirect
13. #3183C2semidirect
14. #3181C2semidirect
15. #3117C2semidirect
16. #3116C2semidirect
17. #3113C2semidirect
18. #3111C2semidirect
19. #2923C2semidirect

Maximal subgroups

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

subgroupindex
#34612
#32632
#32612
#32582
#32442
#32422
#32252
#32232
#32042
#32022
#32012
#31852
#31832
#31812
#31172
#31162
#31132
#31112
#29232