Finite Group Databasebeta

← #22994

Subgroups — #22994

Order 256.

496 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#424C211yesmaximal, characteristic, complemented
5#2392C211yesmaximal, characteristic
6#1955C211yesmaximal, characteristic, complemented
7#1939C211yesmaximal, characteristic, complemented
8#1924C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1851C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#436—24no—
25#436—24no—
26#435—24no—
27#424—24no—
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#398C411yescharacteristic
47#397C411yescharacteristic
48#396C411yescharacteristic
49#396C411yescharacteristic
50#394C411yescharacteristic
51#394C411yescharacteristic
52#394C411yescharacteristic
53#392C411yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×D4C2×C411yescharacteristic
64C22×D4C2×C411yescharacteristic
65C22×D4C2×C411yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95#117—22no—
96#117—22no—
97C2×C4⋊C4C2×C411yescharacteristic
98C2×C4⋊C4C2×C411yescharacteristic
99C2×C4⋊C4C2×C411yescharacteristic
100C2×C4⋊C4C2×C411yescharacteristic
101C2×C4⋊C4C2×C411yescharacteristic
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C4⋊C4C2×C411yescharacteristic
104C2×C22⋊C4C2×C411yescharacteristic
105C2×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C22⋊C4C2×C411yescharacteristic
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8D414yescomplemented
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8D414yescomplemented
134#97—22no—
135#97—22no—
136C4×C8—22no—
137C4×C8—22no—
138#95C2×C411yescharacteristic
139#95C2×C411yescharacteristic
140#95C2×C411yescharacteristic
141#95C2×C411yescharacteristic
142#95C2×C411yescharacteristic
143#95C2×C411yescharacteristic
144C24C22×C411yescharacteristic
145C24C22×C411yescharacteristic
146C24C22×C411yescharacteristic
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4C22×C411yescharacteristic
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186M16—44no—
187M16—44no—
188M16—44no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8C2×D412yes—
211C2×C8C2×D412yes—
212C2×C8C2×D412yes—
213C2×C8C2×D412yes—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42C2×D412yes—
240C42#4112yes—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23C4×D412yes—
262C23C23×C411yescharacteristic
263D4—44no—
264D4—44no—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4C4×D412yes—
308C2×C4C4×D412yes—
309C2×C4C4×D412yes—
310C2×C4#13112yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C42−1+411yescharacteristic
317C8—22no—
318C8—22no—
319C8—22no—
320C8—22no—
321C8—41no—
322C8—41no—
323C8—41no—
324C8—41no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×C4×D411yescharacteristic
347C22C2×C4×D411yescharacteristic
348C22#52011yescharacteristic
349C22C2×#13111yescharacteristic
350C22#56811yescharacteristic
351C22#56811yescharacteristic
352C22#56811yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4D4211yescharacteristic
365C4#54711yescharacteristic
366C4#54811yescharacteristic
367C4#55311yescharacteristic
368C2—42no—
369C2—81no—
370C2—81no—
371C2#236111yescharacteristic
372C2#293011yescharacteristic
373C2#293011yescharacteristic
374C2#293311yescharacteristic
375C2#293511yescharacteristic
376C2#299311yescharacteristic
377C2#299511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radical#392C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#436C2semidirect
2. C2×#436C2semidirect
3. C2×#435C2semidirect
4. C2×#424C2semidirect
6. #1955C2semidirect
7. #1939C2semidirect
8. #1924C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1851C2semidirect
14. #1771C2semidirect
15. #1750C2semidirect
125. C4⋊C8D4semidirect
133. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4352
C2×#4242
#23922
#19552
#19392
#19242
#19232
#18702
#18692
#18512
#17712
#17502