Finite Group Databasebeta

← #17381

Subgroups — #17381

Order 256.

723 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3C22×C22⋊C8C211yesmaximal, characteristic, complemented
4#1472C212yesmaximal, complemented
5#1471C212yesmaximal, complemented
6#1468C214yesmaximal, complemented
7#1459C214yesmaximal, complemented
8C22×#41C411yescharacteristic
9C24×C4C2211yescharacteristic
10C22×M16—21no—
11C23×C8—21no—
12#525—21no—
13#525—21no—
14#525—21no—
15#525—21no—
16#525—21no—
17#525—21no—
18#525—21no—
19#525—21no—
20C2×#123C2212yes—
21C2×C22⋊Q8C411yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C411yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120C412yes—
26C2×C4×D4C412yes—
27C2×C4×D4C2212yes—
28C2×#117C411yescharacteristic
29#407—21no—
30#407—21no—
31#407C2212yes—
32#407C2212yes—
33C2×C22⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×C22⋊C8C2212yes—
40C2×C22⋊C8C2212yes—
41#324C2218yescomplemented
42#323C2218yescomplemented
43C25C2×C411yescharacteristic
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—44no—
49C22×Q8C2×C411yescharacteristic
50C22×D4C2×C412yes—
51C22×D4C2×C412yes—
52C22×D4C2×C412yes—
53C22×D4C2×C411yescharacteristic
54C23×C4—44no—
55C23×C4C2×C411yescharacteristic
56C23×C4C2×C412yes—
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—44no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—44no—
78#123—22no—
79#123—22no—
80#123—22no—
81#123—22no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#120—22no—
99#120—22no—
100#120—22no—
101#120—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4—22no—
110#117—21no—
111#117—21no—
112#117—21no—
113#117—21no—
114C2×C4⋊C4C2×C411yescharacteristic
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C4⋊C4C2×C412yes—
117C2×C4⋊C4C2312yes—
118C2×C22⋊C4C2×C412yes—
119C2×C22⋊C4C2×C412yes—
120C2×C22⋊C4C2×C412yes—
121C2×C22⋊C4C2×C412yes—
122C2×C22⋊C4C2312yes—
123C2×C42C2×C412yes—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2314yes—
131C22⋊C8C2314yes—
132C24—24no—
133C24—42no—
134C24—42no—
135C24C22⋊C411yescharacteristic
136C24C22⋊C411yescharacteristic
137C24C22⋊C411yescharacteristic
138C24C22⋊C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141C24C22×C412yes—
142C24C22×C411yescharacteristic
143#41—44no—
144#41—44no—
145#41—44no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—28no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4C22⋊C411yescharacteristic
180C22×C4C22⋊C411yescharacteristic
181C22×C4C22⋊C411yescharacteristic
182C22×C4C22⋊C411yescharacteristic
183C22×C4C22⋊C411yescharacteristic
184C22×C4C22⋊C411yescharacteristic
185C22×C4C22⋊C411yescharacteristic
186C22×C4C22⋊C411yescharacteristic
187C22×C4C22⋊C411yescharacteristic
188C22×C4C22⋊C411yescharacteristic
189C22×C4C22⋊C411yescharacteristic
190C22×C4C22⋊C411yescharacteristic
191C22×C4C22×C412yes—
192C22×C4C22×C411yescharacteristic
193C22×C4C22×C411yescharacteristic
194C22×C4C22×C412yes—
195C22×C4C22×C412yes—
196C22×C4C22×C411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2411yescharacteristic
210M16—44no—
211M16—44no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—44no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C42—22no—
237C42—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—24no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—416no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊C411yescharacteristic
264C23C2×C22⋊C411yescharacteristic
265C23C2×C22⋊C411yescharacteristic
266C23C2×C22⋊C411yescharacteristic
267C23C2×C22⋊C411yescharacteristic
268C23C2×C22⋊C411yescharacteristic
269C23#13114yes—
270C23C23×C411yescharacteristic
271Q8—42no—
272Q8—42no—
273D4—42no—
274D4—44no—
275D4—44no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—42no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4#9914yes—
317C2×C4C2×C22⋊C411yescharacteristic
318C2×C4C2×C22⋊C411yescharacteristic
319C2×C4C2×C22⋊C411yescharacteristic
320C2×C4C2×C22⋊C411yescharacteristic
321C2×C4C2×C22⋊C411yescharacteristic
322C2×C4C2×C22⋊C411yescharacteristic
323C2×C4C22×D411yescharacteristic
324C2×C4C22×D411yescharacteristic
325C8—44no—
326C8—44no—
327C22—28no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—84no—
354C22#40812yes—
355C22#40812yes—
356C22C2×#9912yes—
357C22#41312yes—
358C22C22×C22⋊C411yescharacteristic
359C22C2×#13112yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—42no—
369C4C2×#9912yes—
370C4C2×#9912yes—
371C2—22no—
372C2—22no—
373C2—22no—
374C2—22no—
375C2—22no—
376C2—41no—
377C2—41no—
378C2—82no—
379C2C2×#40811yescharacteristic
380C2C22×#9911yescharacteristic
381C2C2×#41311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radicalC2×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. C22×C22⋊C8C2semidirect
4. #1472C2semidirect
5. #1471C2semidirect
6. #1468C2semidirect
7. #1459C2semidirect
41. #324C22semidirect
42. #323C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4072
C22×C22⋊C82
#14722
#14712
#14682
#14592