Finite Group Databasebeta

← #17609

Subgroups — #17609

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3495C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic, complemented
3#2343C211yesmaximal, characteristic, complemented
4#1521C211yesmaximal, characteristic, complemented
5#1518C211yesmaximal, characteristic, complemented
6#1516C211yesmaximal, characteristic
7#1515C211yesmaximal, characteristic, complemented
8#1512C211yesmaximal, characteristic, complemented
9#1511C211yesmaximal, characteristic, complemented
10#1509C211yesmaximal, characteristic, complemented
11#1506C211yesmaximal, characteristic, complemented
12#1504C211yesmaximal, characteristic
13#1503C211yesmaximal, characteristic, complemented
14#1500C211yesmaximal, characteristic, complemented
15C2×#328C211yesmaximal, characteristic, complemented
16#556—21no—
17#554—21no—
18#552—21no—
19#552—21no—
20#552—21no—
21#552—21no—
22#551—21no—
23#551—21no—
24#544C411yescharacteristic, complemented
25#542C411yescharacteristic, complemented
26#542C411yescharacteristic, complemented
27#541C411yescharacteristic, complemented
28#541C411yescharacteristic, complemented
29#536C411yescharacteristic, complemented
30#533C2211yescharacteristic
31#533C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#525C2211yescharacteristic
35C2×C22⋊Q8C2211yescharacteristic
36#520C411yescharacteristic, complemented
37#519C411yescharacteristic, complemented
38C2×#117C2211yescharacteristic
39#433—21no—
40#433—21no—
41#432C2211yescharacteristic
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44#431C2211yescharacteristic
45#424—21no—
46#407C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#405—21no—
49#387—21no—
50#387—21no—
51#385—21no—
52#383C2211yescharacteristic
53#382C2211yescharacteristic
54#381C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C22⋊C4—21no—
57#376C2211yescharacteristic
58C2×#95C2211yescharacteristic
59#328C2211yescharacteristic
60#328C2211yescharacteristic
61#328C2211yescharacteristic
62#328C2211yescharacteristic
63#328C2211yescharacteristic
64#328C2211yescharacteristic
65#328C2211yescharacteristic
66#328C2211yescharacteristic
67#328C2211yescharacteristic
68#328C2211yescharacteristic
69#328C2211yescharacteristic
70#328C2211yescharacteristic
71#327C2211yescharacteristic
72#327C2211yescharacteristic
73#327C2211yescharacteristic
74#327C2211yescharacteristic
75C2×#41C2×C411yescharacteristic
76C22×Q8C2×C411yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×M16—21no—
80C22×C8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2×C411yescharacteristic, complemented
83C4⋊Q8C2×C411yescharacteristic, complemented
84C4⋊Q8C2×C411yescharacteristic, complemented
85C4⋊Q8C2×C411yescharacteristic, complemented
86#126—21no—
87#126—21no—
88#126—21no—
89#126—21no—
90#126—21no—
91#126—21no—
92#126—21no—
93#126—21no—
94#125—21no—
95#125—21no—
96#125—21no—
97#125—21no—
98#125—21no—
99#125—21no—
100#125—21no—
101#125C2×C411yescharacteristic, complemented
102#125C2×C411yescharacteristic, complemented
103#124—21no—
104#124—21no—
105#124C2×C411yescharacteristic, complemented
106#124C2×C411yescharacteristic, complemented
107#123—21no—
108#123—21no—
109#123—21no—
110#123—21no—
111#123—21no—
112#123—21no—
113#123—21no—
114C22⋊Q8—21no—
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8C2×C411yescharacteristic
125C22⋊Q8C2×C411yescharacteristic
126C22⋊Q8C2×C411yescharacteristic
127C22⋊Q8C2311yescharacteristic
128C22⋊Q8C2311yescharacteristic
129C22⋊Q8C2311yescharacteristic
130#121C2×C411yescharacteristic
131#121C2311yescharacteristic
132#120—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8C2×C411yescharacteristic, complemented
138C4×Q8C2×C411yescharacteristic, complemented
139C4×Q8C2×C411yescharacteristic, complemented
140C4×Q8C2×C411yescharacteristic, complemented
141C4×Q8C2×C411yescharacteristic, complemented
142C4×Q8C2×C411yescharacteristic, complemented
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4C2×C411yescharacteristic, complemented
150C4×D4C2×C411yescharacteristic, complemented
151#117—21no—
152#117—21no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157#117—21no—
158#117C2×C411yescharacteristic
159#117C2×C411yescharacteristic
160#117C2×C411yescharacteristic
161#117C2×C411yescharacteristic
162#117C2×C411yescharacteristic
163#117D411yescharacteristic
164#117D411yescharacteristic
165#117D411yescharacteristic
166#117D411yescharacteristic
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C2×C411yescharacteristic
172C2×C4⋊C4D411yescharacteristic
173C2×C4⋊C4D411yescharacteristic
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
206C24C2×D411yescharacteristic
207#41—42no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—42no—
212C2×Q8C22×C411yescharacteristic
213C2×Q8C22×C411yescharacteristic
214C2×Q8C22×C411yescharacteristic
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4C22×C411yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4C22×C411yescharacteristic
233C22×C4C22×C411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2411yescharacteristic
242M16—42no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—42no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
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—42no—
272C4⋊C4—81no—
273C4⋊C4C22⋊C411yescharacteristic
274C4⋊C4C22⋊C411yescharacteristic
275C4⋊C4C22⋊C411yescharacteristic
276C4⋊C4C22⋊C411yescharacteristic
277C4⋊C4C22⋊C411yescharacteristic
278C4⋊C4C22⋊C411yescharacteristic
279C4⋊C4C22⋊C411yescharacteristic
280C4⋊C4C22⋊C411yescharacteristic
281C4⋊C4C22×C411yescharacteristic
282C4⋊C4C22×C411yescharacteristic
283C4⋊C4C22×C411yescharacteristic
284C4⋊C4C22×C411yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
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—42no—
301C22⋊C4C22⋊C411yescharacteristic
302C22⋊C4C22⋊C411yescharacteristic
303C22⋊C4C22⋊C411yescharacteristic
304C22⋊C4C22⋊C411yescharacteristic
305C22⋊C4C22⋊C411yescharacteristic
306C22⋊C4C22⋊C411yescharacteristic
307C22⋊C4C22⋊C411yescharacteristic
308C22⋊C4C22⋊C411yescharacteristic
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—81no—
318C42C22×C411yescharacteristic
319C42C22×C411yescharacteristic
320C42C22×C411yescharacteristic
321C42C22×C411yescharacteristic
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2×D411yescharacteristic
325C42C2×D411yescharacteristic
326C23—21no—
327C23—21no—
328C23—22no—
329C23—21no—
330C23—41no—
331C23C2×C22⋊C411yescharacteristic
332C23C2×C22⋊C411yescharacteristic
333C23C22×D411yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—84no—
382C2×C4C2×C22⋊C411yescharacteristic
383C2×C4C2×C22⋊C411yescharacteristic
384C2×C4C2×C22⋊C411yescharacteristic
385C2×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22#41012yes—
411C22C22×C22⋊C411yescharacteristic
412C22#57711yescharacteristic
413C22#57811yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—81no—
435C4—81no—
436C4—81no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—41no—
441C2—81no—
442C2C2×#41011yescharacteristic
443C2#289811yescharacteristic
444C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#525C22
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. #1518C2semidirect
7. #1515C2semidirect
8. #1512C2semidirect
9. #1511C2semidirect
10. #1509C2semidirect
11. #1506C2semidirect
13. #1503C2semidirect
14. #1500C2semidirect
15. C2×#328C2semidirect
24. #544C4semidirect
25. #542C4semidirect
26. #542C4semidirect
27. #541C4semidirect
28. #541C4semidirect
29. #536C4semidirect
36. #520C4semidirect
37. #519C4semidirect
82. C4⋊Q8C2×C4semidirect
83. C4⋊Q8C2×C4semidirect
84. C4⋊Q8C2×C4semidirect
85. C4⋊Q8C2×C4semidirect
101. #125C2×C4semidirect
102. #125C2×C4semidirect
105. #124C2×C4semidirect
106. #124C2×C4semidirect
137. C4×Q8C2×C4semidirect
138. C4×Q8C2×C4semidirect
139. C4×Q8C2×C4semidirect
140. C4×Q8C2×C4semidirect
141. C4×Q8C2×C4semidirect
142. C4×Q8C2×C4semidirect
149. C4×D4C2×C4semidirect
150. C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34952
#29242
#23432
#15212
#15182
#15162
#15152
#15122
#15112
#15092
#15062
#15042
#15032
#15002
C2×#3282