Finite Group Databasebeta

← #17680

Subgroups — #17680

Order 256.

439 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515C211yesmaximal, characteristic
2#2970C211yesmaximal, characteristic, complemented
3#2923C211yesmaximal, characteristic, complemented
4#1537C211yesmaximal, characteristic, complemented
5#1535C211yesmaximal, characteristic, complemented
6#1535C211yesmaximal, characteristic, complemented
7#1534C211yesmaximal, characteristic, complemented
8#1532C211yesmaximal, characteristic, complemented
9#1532C211yesmaximal, characteristic, complemented
10#1531C211yesmaximal, characteristic, complemented
11#1529C211yesmaximal, characteristic, complemented
12#1528C211yesmaximal, characteristic, complemented
13#1528C211yesmaximal, characteristic, complemented
14#1524C211yesmaximal, characteristic, complemented
15C2×#330C211yesmaximal, characteristic, complemented
16#557C411yescharacteristic
17#554C411yescharacteristic
18#554C411yescharacteristic
19#553C411yescharacteristic
20#553C411yescharacteristic
21#552—21no—
22#552—21no—
23#552—21no—
24#552C411yescharacteristic
25#552C411yescharacteristic
26#547—21no—
27#547C411yescharacteristic
28#541—21no—
29#539—21no—
30#539—21no—
31#533C2211yescharacteristic
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#529C2211yescharacteristic
35C2×#125C2211yescharacteristic
36C2×#125C2211yescharacteristic
37#520—21no—
38C4×#41C2211yescharacteristic
39#446—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic
46#432C2211yescharacteristic
47#432C2211yescharacteristic
48#432C2211yescharacteristic
49#432C2211yescharacteristic
50#432C2211yescharacteristic
51#432C2211yescharacteristic
52#424—21no—
53#408—21no—
54C4×M16C2211yescharacteristic
55C4×M16C2211yescharacteristic
56C2×#97C2211yescharacteristic
57C2×#97C2211yescharacteristic
58C2×#97C2211yescharacteristic
59#330C2211yescharacteristic
60#330C2211yescharacteristic
61#330C2211yescharacteristic
62#330C2211yescharacteristic
63#330C2211yescharacteristic
64#330C2211yescharacteristic
65#330C2211yescharacteristic
66#330C2211yescharacteristic
67#330C2211yescharacteristic
68#330C2211yescharacteristic
69#330C2211yescharacteristic
70#330C2211yescharacteristic
71#329C2211yescharacteristic
72#329C2211yescharacteristic
73#329C2211yescharacteristic
74#329C2211yescharacteristic
75C2×#41—21no—
76C2×#41D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2×C411yescharacteristic
84C4⋊Q8C2×C411yescharacteristic
85#126—21no—
86#126—21no—
87#126—21no—
88#126—21no—
89#126—21no—
90#126—21no—
91#126—21no—
92#125—21no—
93#125—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
102#125C2×C411yescharacteristic
103#125C2×C411yescharacteristic
104#125C2×C411yescharacteristic
105#125C2311yescharacteristic
106#125C2311yescharacteristic
107#125C2311yescharacteristic
108#124C2311yescharacteristic
109#123—21no—
110#123—21no—
111#123—21no—
112#123—21no—
113#123—21no—
114#123—21no—
115#123—21no—
116#123—21no—
117#123—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125C22⋊Q8—21no—
126C22⋊Q8C2×C411yescharacteristic
127C22⋊Q8C2×C411yescharacteristic
128C22⋊Q8C2×C411yescharacteristic
129#121—21no—
130#121—21no—
131#121C2×C411yescharacteristic
132#121C2×C411yescharacteristic
133#121C2×C411yescharacteristic
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8C2×C411yescharacteristic
138C4×Q8C2×C411yescharacteristic
139C4×Q8C2×C411yescharacteristic
140C4×Q8C2×C411yescharacteristic
141C4×Q8D411yescharacteristic
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4C2×C411yescharacteristic
149C4×D4C2×C411yescharacteristic
150C4×D4C2×C411yescharacteristic
151C4×D4C2×C411yescharacteristic
152C4×D4C2×C411yescharacteristic
153C4×D4C2×C411yescharacteristic
154C4×D4D411yescharacteristic
155C4×D4D411yescharacteristic
156C4×D4D411yescharacteristic
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—21no—
162#117C2×C411yescharacteristic
163#117C2×C411yescharacteristic
164#117D411yescharacteristic
165#117D411yescharacteristic
166#117D411yescharacteristic
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C411yescharacteristic
173C2×C4⋊C4C2×C411yescharacteristic
174C2×C4⋊C4C2×C411yescharacteristic
175C2×C4⋊C4C2×C411yescharacteristic
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—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193#97—21no—
194#97—21no—
195#97—21no—
196#97C2311yescharacteristic
197#97C2311yescharacteristic
198#97C2311yescharacteristic
199#97C2311yescharacteristic
200#97C2311yescharacteristic
201#97C2311yescharacteristic
202#97C2311yescharacteristic
203#97C2311yescharacteristic
204C4×C8—21no—
205C4×C8—21no—
206#41—42no—
207#41—44no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8C22⋊C411yescharacteristic
211C2×Q8C22×C411yescharacteristic
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4C22⋊C411yescharacteristic
216C2×D4C22⋊C411yescharacteristic
217C2×D4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
227C22×C4C22×C411yescharacteristic
228C22×C4C22×C411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—42no—
246C2×C8—42no—
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—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
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
285C4⋊C4C22×C411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
300C22⋊C4C22⋊C411yescharacteristic
301C22⋊C4C22⋊C411yescharacteristic
302C22⋊C4C22⋊C411yescharacteristic
303C22⋊C4C22⋊C411yescharacteristic
304C22⋊C4C22⋊C411yescharacteristic
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42C22×C411yescharacteristic
311C42C22×C411yescharacteristic
312C42C22×C411yescharacteristic
313C42C2×D411yescharacteristic
314C42C2×D411yescharacteristic
315C42C2×D411yescharacteristic
316C42C2×D411yescharacteristic
317C42C2×D411yescharacteristic
318C42C2×D411yescharacteristic
319C42C2411yescharacteristic
320C23—21no—
321C23C2×C22⋊C411yescharacteristic
322C23C2×C22⋊C411yescharacteristic
323C23C2×C22⋊C411yescharacteristic
324Q8—41no—
325Q8—41no—
326Q8—42no—
327Q8—81no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—42no—
333D4—42no—
334D4—81no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C2×C22⋊C411yescharacteristic
359C2×C4C2×C22⋊C411yescharacteristic
360C2×C4C2×C22⋊C411yescharacteristic
361C2×C4C2×C22⋊C411yescharacteristic
362C2×C4C2×C22⋊C411yescharacteristic
363C2×C4C2×C22⋊C411yescharacteristic
364C2×C4C2×C22⋊C411yescharacteristic
365C2×C4C2×C22⋊C411yescharacteristic
366C2×C4C2×C22⋊C411yescharacteristic
367C2×C4C23×C411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C8—22no—
371C8—41no—
372C8—41no—
373C8—41no—
374C8—41no—
375C8—41no—
376C8—41no—
377C8—41no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22#42112yes—
390C22C22×C22⋊C411yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C2—22no—
411C2—41no—
412C2—41no—
413C2—81no—
414C2#288911yescharacteristic
415C2C2×#42111yescharacteristic
416C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3515C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2970C2semidirect
3. #2923C2semidirect
4. #1537C2semidirect
5. #1535C2semidirect
6. #1535C2semidirect
7. #1534C2semidirect
8. #1532C2semidirect
9. #1532C2semidirect
10. #1531C2semidirect
11. #1529C2semidirect
12. #1528C2semidirect
13. #1528C2semidirect
14. #1524C2semidirect
15. C2×#330C2semidirect

Maximal subgroups

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

subgroupindex
#35152
#29702
#29232
#15372
#15352
#15342
#15322
#15312
#15292
#15282
#15242
C2×#3302