Finite Group Databasebeta

← #31198

Subgroups — #31198

Order 256.

478 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3#2416C211yesmaximal, characteristic, complemented
4#2414C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1999C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1955C211yesmaximal, characteristic, complemented
13#1943C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#121—21no—
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#482—24no—
26#480—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44#396—21no—
45#394—21no—
46#394—21no—
47#392—21no—
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#386—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4D411yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×D8—22no—
72C2×D8—41no—
73C2×D8—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#121—22no—
79#121—22no—
80#121—41no—
81#121—41no—
82#121—41no—
83#121—41no—
84#121—41no—
85#121—41no—
86#121—44no—
87C4×D4—41no—
88C4×D4—41no—
89#117—22no—
90#117—41no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118#107—22no—
119#107D414yescomplemented
120#106—22no—
121#106—41no—
122#106—41no—
123C4⋊C8—41no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134#97—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2411yescharacteristic
196D8—44no—
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×C8C2×D412yes—
208C2×C8#4112yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—81no—
223C4⋊C4C2×D412yes—
224C4⋊C4C2×D412yes—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—21no—
239C42—21no—
240C42—42no—
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—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23AGL(1,ℤ/8)12yes—
269C23C22×D411yescharacteristic
270D4—44no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—84no—
280D4—84no—
281D4—84no—
282D4—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
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×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C2×C42−1+411yescharacteristic
328C8—22no—
329C8—22no—
330C8—41no—
331C8—41no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#53411yescharacteristic
361C22D4211yescharacteristic
362C22#54711yescharacteristic
363C22C2×AGL(1,ℤ/8)11yescharacteristic
364C22#57511yescharacteristic
365C22#57611yescharacteristic
366C22#57711yescharacteristic
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—42no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4#53711yescharacteristic
378C4#53811yescharacteristic
379C4#55011yescharacteristic
380C4#56211yescharacteristic
381C2—42no—
382C2—81no—
383C2—161no—
384C2—161no—
385C2#268211yescharacteristic
386C2#319411yescharacteristic
387C2#319811yescharacteristic
388C2#326411yescharacteristic
389C2#327111yescharacteristic
390C2#334511yescharacteristic
391C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C2×#480C2semidirect
3. #2416C2semidirect
4. #2414C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #1999C2semidirect
10. #1999C2semidirect
11. #1972C2semidirect
12. #1955C2semidirect
13. #1943C2semidirect
14. #1854C2semidirect
15. #1853C2semidirect
119. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4802
#24162
#24142
#20672
#20652
#20342
#19992
#19722
#19552
#19432
#18542
#18532