Finite Group Databasebeta

← #61966

Subgroups — #61966

Order 256.

912 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic
2Q8×#41C211yesmaximal, characteristic, complemented
3#3398C212yesmaximal, complemented
4#3369C212yesmaximal, complemented
5#3331C212yesmaximal, complemented
6#3316C214yesmaximal, complemented
7#3276C212yesmaximal, complemented
8#3270C214yesmaximal, complemented
9#3255C212yesmaximal, complemented
10#3246C211yesmaximal, characteristic, complemented
11#3216C212yesmaximal, complemented
12#3209C211yesmaximal, characteristic, complemented
13#3113C212yesmaximal, complemented
14#3103C211yesmaximal, characteristic, complemented
15#3037C212yesmaximal, complemented
16C2×#462C211yesmaximal, characteristic, complemented
17#2989C211yesmaximal, characteristic, complemented
18C2×#137C2214yes—
19C22×Q16C2212yes—
20Q82C2211yescharacteristic
21#557—21no—
22#557C2211yescharacteristic
23#556C2212yes—
24#555—21no—
25#554—21no—
26#554—22no—
27#554—21no—
28#554C2211yescharacteristic
29#554C2212yes—
30#553—22no—
31D4×Q8C2212yes—
32D4×Q8C2211yescharacteristic
33#548C2214yes—
34#547C2212yes—
35#542—22no—
36#537—21no—
37#533—21no—
38#533—22no—
39#533C2212yes—
40#533C2211yescharacteristic
41#533C2212yes—
42#532C2212yes—
43#532C2211yescharacteristic
44C2×C4⋊Q8C2211yescharacteristic
45#520—21no—
46C4×#41—21no—
47C4×#41C2211yescharacteristic
48C2×C4×Q8C2211yescharacteristic
49C2×C4×Q8C2212yes—
50#497C2214yes—
51#494C2212yes—
52#489C2214yes—
53#487C2212yes—
54#483C2214yes—
55#481C2212yes—
56#476C2212yes—
57#475C2211yescharacteristic
58#475C2211yescharacteristic
59#475C2212yes—
60#475C2214yes—
61#470C2214yes—
62#470C2214yes—
63#465C2214yes—
64#464C2214yes—
65#464C2214yes—
66#462C2212yes—
67#462C2212yes—
68#462C2212yes—
69#462C2212yes—
70#462C2214yes—
71#462C2212yes—
72#462C2212yes—
73#461C2212yes—
74#460C2211yescharacteristic
75#460C2211yescharacteristic
76#460C2212yes—
77#452C2214yes—
78#452C2214yes—
79#451C2214yes—
80#451C2214yes—
81#448C2212yes—
82#446C2212yes—
83#441C2214yes—
84#441C2214yes—
85C4×SD16C2212yes—
86C4×SD16C2212yes—
87#436C2212yes—
88#435C2214yes—
89#432C2212yes—
90#431C2211yescharacteristic
91#423C2212yes—
92C2×C4⋊C8C2211yescharacteristic
93#417C2214yes—
94C2×#103C2212yes—
95C2×#41—21no—
96C2×#41D411yescharacteristic
97C2×#41C2314yes—
98C22×Q8C2311yescharacteristic
99C22×Q8C2312yes—
100C22×Q8C2312yes—
101#137—48no—
102C2×Q16—44no—
103C2×Q16C2314yes—
104C2×Q16C2314yes—
105C2×Q16C2314yes—
106C2×Q16C2314yes—
107C2×SD16C2312yes—
108C2×SD16C2314yes—
109C2×SD16C2312yes—
110C2×M16C2314yes—
111C22×C8C2312yes—
112C4⋊Q8—21no—
113C4⋊Q8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C2312yes—
119C4⋊Q8C2312yes—
120C4⋊Q8C2311yescharacteristic
121C4⋊Q8C2312yes—
122C4⋊Q8C2311yescharacteristic
123C4⋊Q8C2312yes—
124C4⋊Q8C2311yescharacteristic
125#127C2311yescharacteristic
126#126—24no—
127#125—21no—
128#125—22no—
129#125—21no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—21no—
134#124—24no—
135#124—21no—
136#124C2314yes—
137#123—24no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—24no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8C2314yes—
150C22⋊Q8C2312yes—
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153#121—22no—
154#121—24no—
155#121—22no—
156#121—21no—
157#121—22no—
158#121C2312yes—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—22no—
164C4×Q8—21no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8D411yescharacteristic
169C4×Q8C2311yescharacteristic
170C4×Q8C2311yescharacteristic
171C4×Q8C2312yes—
172C4×Q8C2311yescharacteristic
173C4×Q8C2312yes—
174C4×Q8C2311yescharacteristic
175C4×Q8C2312yes—
176C4×Q8C2314yes—
177C4×Q8C2311yescharacteristic
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—22no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4D412yes—
187C4×D4D411yescharacteristic
188C4×D4C2312yes—
189C4×D4C2311yescharacteristic
190C4×D4C2312yes—
191#117—22no—
192#117—21no—
193#117—22no—
194#117—22no—
195#117D411yescharacteristic
196#117D412yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C2311yescharacteristic
203C2×C4⋊C4C2312yes—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42C2311yescharacteristic
207C2×C42C2312yes—
208#106C2312yes—
209#106C2312yes—
210#106C2314yes—
211C4⋊C8C2312yes—
212C4⋊C8C2312yes—
213C4⋊C8C2311yescharacteristic
214C4⋊C8C2314yes—
215C4⋊C8C2311yescharacteristic
216#103C2314yes—
217#103C2314yes—
218#103C2314yes—
219#103C2312yes—
220#103C2312yes—
221#103C2314yes—
222#103C2314yes—
223#102C2312yes—
224#102C2314yes—
225#102C2312yes—
226C22⋊C8C2312yes—
227C22⋊C8C2314yes—
228#97C2314yes—
229C4×C8C2312yes—
230#41—24no—
231#41—44no—
232#41—44no—
233#41—44no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8C2×D411yescharacteristic
247C2×Q8C2414yes—
248C2×Q8C2412yes—
249C2×Q8C2411yescharacteristic
250C2×Q8C2411yescharacteristic
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4C2×D412yes—
256C2×D4C2×D411yescharacteristic
257C2×D4C2412yes—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4C2×D412yes—
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D412yes—
268C22×C4C2×D411yescharacteristic
269C22×C4C2411yescharacteristic
270C22×C4C2412yes—
271Q16—28no—
272Q16—44no—
273Q16—44no—
274Q16—44no—
275SD16—24no—
276SD16—42no—
277SD16—44no—
278M16—44no—
279C2×C8—42no—
280C2×C8C2412yes—
281C2×C8C2412yes—
282C2×C8C2414yes—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D412yes—
311C4⋊C4C2×D412yes—
312C4⋊C4C2411yescharacteristic
313C4⋊C4C2412yes—
314C4⋊C4C2411yescharacteristic
315C4⋊C4C2412yes—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C42—21no—
328C42—22no—
329C42—21no—
330C42—22no—
331C42—21no—
332C42—22no—
333C42—21no—
334C42—24no—
335C42C2×D412yes—
336C42C2×D411yescharacteristic
337C42C2×D412yes—
338C42C2×D411yescharacteristic
339C42C2411yescharacteristic
340C42C2412yes—
341C42C2411yescharacteristic
342C23—22no—
343C23C22×D412yes—
344C23C22×D411yescharacteristic
345Q8—22no—
346Q8—21no—
347Q8—22no—
348Q8—24no—
349Q8—21no—
350Q8—21no—
351Q8—22no—
352Q8—24no—
353Q8—44no—
354Q8—42no—
355Q8—42no—
356Q8—82no—
357Q8—81no—
358Q8—82no—
359Q8H(5,𝔽2)12yes—
360D4—24no—
361D4—22no—
362D4—22no—
363D4—22no—
364D4—42no—
365D4—44no—
366D4—42no—
367D4—84no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#13714yescomplemented
391C2×C4C22×D412yes—
392C2×C4C22×D412yes—
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D412yes—
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C2×C4C2511yescharacteristic
400C8—24no—
401C8—22no—
402C8—22no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22C2×#13712yes—
412C22C23×D411yescharacteristic
413C4—21no—
414C4—22no—
415C4—21no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—42no—
423C4—42no—
424C4—41no—
425C4—44no—
426C4—42no—
427C4C2×#13712yes—
428C4C2×#13712yes—
429C4C2×H(5,𝔽2)11yescharacteristic
430C4#58511yescharacteristic
431C2—22no—
432C2—42no—
433C2—82no—
434C2#348811yescharacteristic
435C2C22×#13711yescharacteristic
436C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3511C2
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
2. Q8×#41C2semidirect
3. #3398C2semidirect
4. #3369C2semidirect
5. #3331C2semidirect
6. #3316C2semidirect
7. #3276C2semidirect
8. #3270C2semidirect
9. #3255C2semidirect
10. #3246C2semidirect
11. #3216C2semidirect
12. #3209C2semidirect
13. #3113C2semidirect
14. #3103C2semidirect
15. #3037C2semidirect
16. C2×#462C2semidirect
17. #2989C2semidirect
390. C2×C4#137semidirect

Maximal subgroups

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

subgroupindex
#35112
Q8×#412
#33982
#33692
#33312
#33162
#32762
#32702
#32552
#32462
#32162
#32092
#31132
#31032
#30372
C2×#4622
#29892