Finite Group Databasebeta

← #16660

Subgroups — #16660

Order 256.

1019 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#392C211yesmaximal, characteristic
2C2×#386C212yesmaximal, complemented
3C2×#381C211yesmaximal, characteristic, complemented
4#1443C212yesmaximal, complemented
5#1440C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1439C211yesmaximal, characteristic, complemented
8#1438C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10C23×D4C2212yes—
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22#392—24no—
23#386—28no—
24#381—24no—
25C2×#95C2211yescharacteristic
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C25C2312yes—
35C25C2311yescharacteristic
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—24no—
39C22×D4—24no—
40C22×D4—24no—
41C22×D4—22no—
42C22×D4—24no—
43C22×D4—42no—
44C22×D4—42no—
45C22×D4—42no—
46C22×D4—41no—
47C22×D4—41no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—41no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4—41no—
54C23×C4D414yescomplemented
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2312yes—
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—42no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4D414yes—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—24no—
92#95—22no—
93#95—24no—
94#95—41no—
95#95—42no—
96#95—42no—
97C24—22no—
98C24—22no—
99C24—21no—
100C24—21no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24C2×D412yes—
120C24C2×D412yes—
121C24#4114yes—
122C24#4112yes—
123C24C2411yescharacteristic
124C2×D4—216no—
125C2×D4—48no—
126C2×D4—48no—
127C2×D4—48no—
128C2×D4—44no—
129C2×D4—48no—
130C2×D4—48no—
131C2×D4—44no—
132C2×D4—48no—
133C2×D4—48no—
134C2×D4—44no—
135C2×D4—48no—
136C22×C4—28no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D414yes—
187C22×C4#4114yes—
188C4⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—44no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C23—24no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23#12114yes—
255C23C22×D411yescharacteristic
256C23C22×D411yescharacteristic
257C23C2×#4111yescharacteristic
258C23C2×#4112yes—
259C23C2×#4112yes—
260C23H(5,𝔽2)11yescharacteristic
261C23H(5,𝔽2)11yescharacteristic
262C23H(5,𝔽2)11yescharacteristic
263C23H(5,𝔽2)11yescharacteristic
264C23H(5,𝔽2)12yes—
265C23H(5,𝔽2)12yes—
266D4—432no—
267D4—416no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—416no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C22—24no—
315C22—28no—
316C22—42no—
317C22—44no—
318C22—42no—
319C22—44no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22C2×#12111yescharacteristic
341C22#52512yes—
342C22#53411yescharacteristic
343C22#53411yescharacteristic
344C22#53411yescharacteristic
345C22#53511yescharacteristic
346C22#53512yes—
347C22#53811yescharacteristic
348C22#53811yescharacteristic
349C22#53812yes—
350C22#53812yes—
351C22#54012yes—
352C22#54012yes—
353C22#54612yes—
354C22#54612yes—
355C22#55111yescharacteristic
356C22#55112yes—
357C22#55112yes—
358C22#55312yes—
359C22#55911yescharacteristic
360C22#56011yescharacteristic
361C22#56012yes—
362C22#56011yescharacteristic
363C4—44no—
364C4—44no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—82no—
370C2—44no—
371C2—84no—
372C2#241112yes—
373C2#260011yescharacteristic
374C2#261711yescharacteristic
375C2#263512yes—
376C2#269711yescharacteristic
377C2#270312yes—
378C2#270911yescharacteristic
379C2#274912yes—
380C2#275211yescharacteristic
381C2#276011yescharacteristic
382C2#281511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC2×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#386C2semidirect
3. C2×#381C2semidirect
4. #1443C2semidirect
5. #1440C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
54. C23×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
C2×#3862
C2×#3812
#14432
#14402
#14392
#14382