Finite Group Databasebeta

← #22860

Subgroups — #22860

Order 256.

587 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic, complemented
6#1973C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#449C418yescomplemented
26#442—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29#416—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390—21no—
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#386—21no—
48#386—21no—
49#386—21no—
50#381—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41C2×C414yescomplemented
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16C2×C414yescomplemented
81C2×D8—22no—
82C2×D8—22no—
83C2×D8C2×C414yes—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91#121—22no—
92#121C2×C414yescomplemented
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99#117—22no—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—41no—
122#106—22no—
123#103—22no—
124#103—22no—
125#103C2×C414yescomplemented
126#102—22no—
127#102—22no—
128#102—41no—
129#102C2×C414yescomplemented
130#102D414yescomplemented
131C22⋊C8C2×C414yescomplemented
132#97—22no—
133#97—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95D411yescharacteristic
142#95D411yescharacteristic
143C24—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146#41—28no—
147#41—44no—
148#41—44no—
149#41—44no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C22×C412yes—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4C22×C412yes—
171C2×D4C22×C412yes—
172C2×D4C2×D412yes—
173C2×D4#4112yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4C22×C412yes—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4#4112yes—
203C22×C4C2411yescharacteristic
204SD16—44no—
205D8—28no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—44no—
216C2×C8C22×C412yes—
217C2×C8C22×C412yes—
218C2×C8C2×D412yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—81no—
228C4⋊C4C22×C412yes—
229C4⋊C4C2×D412yes—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—81no—
246C23—22no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C4×D412yes—
267C23C22×D411yescharacteristic
268Q8—22no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—22no—
277D4—48no—
278D4—41no—
279D4—41no—
280D4—84no—
281D4C4×D414yes—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—48no—
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—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4C4×D412yes—
323C2×C4C23×C411yescharacteristic
324C2×C4C22×D411yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C8—22no—
330C8—22no—
331C8—22no—
332C8—41no—
333C22—22no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22C2×C4×D411yescharacteristic
355C22D4211yescharacteristic
356C22#54611yescharacteristic
357C22#57511yescharacteristic
358C22#57511yescharacteristic
359C22#57611yescharacteristic
360C22#57711yescharacteristic
361C4—22no—
362C4—42no—
363C4—42no—
364C4—41no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4C2×C4×D411yescharacteristic
373C4#51811yescharacteristic
374C4#54611yescharacteristic
375C4#55111yescharacteristic
376C2—42no—
377C2—44no—
378C2—82no—
379C2#235811yescharacteristic
380C2#294311yescharacteristic
381C2#294611yescharacteristic
382C2#326711yescharacteristic
383C2#326811yescharacteristic
384C2#330711yescharacteristic
385C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#442C2semidirect
3. C2×#416C2semidirect
4. #2363C2semidirect
5. #2339C2semidirect
6. #1973C2semidirect
7. #1935C2semidirect
8. #1927C2semidirect
9. #1893C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1862C2semidirect
13. #1854C2semidirect
14. #1762C2semidirect
15. #1752C2semidirect
25. #449C4semidirect
67. C2×#41C2×C4semidirect
80. C2×SD16C2×C4semidirect
92. #121C2×C4semidirect
125. #103C2×C4semidirect
129. #102C2×C4semidirect
130. #102D4semidirect
131. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4422
C2×#4162
#23632
#23392
#19732
#19352
#19272
#18932
#18792
#18772
#18622
#18542
#17622
#17522