Finite Group Databasebeta

← #22056

Subgroups — #22056

Order 256.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2#2324C211yesmaximal, characteristic, complemented
3#2309C211yesmaximal, characteristic, complemented
4#1972C211yesmaximal, characteristic, complemented
5#1970C211yesmaximal, characteristic, complemented
6#1936C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#447C418yescomplemented
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic, complemented
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic, complemented
37C2×C4×C8C2211yescharacteristic, complemented
38#390—21no—
39#390—21no—
40#390—21no—
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
47#388—21no—
48#386—21no—
49#385—21no—
50#380C2211yescharacteristic
51#379—21no—
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60C25—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—22no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4C2×C414yescomplemented
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×D8—41no—
75C2×D8—41no—
76C2×D8—41no—
77C2×D8—41no—
78C2×D8C2×C414yes—
79C2×D8C2×C414yes—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#121—22no—
85#121C2×C414yes—
86C4×D4—41no—
87C4×D4—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91#117—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
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×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—41no—
121#106—41no—
122#102—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126#102—41no—
127#102—41no—
128#102C2×C414yescomplemented
129#102C2×C414yescomplemented
130C22⋊C8—22no—
131C22⋊C8C2×C414yescomplemented
132#97—41no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—24no—
141C24—42no—
142C24—42no—
143C24—81no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24C2×D411yescharacteristic
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4C22×C412yes—
174C2×D4C22×C412yes—
175C2×D4C22×C412yes—
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—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4C22×C412yes—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4#4112yes—
195C22×C4C2411yescharacteristic
196D8—44no—
197D8—44no—
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×C8C22×C412yes—
209C2×C8C22×C412yes—
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—41no—
221C4⋊C4—81no—
222C4⋊C4C22×C412yes—
223C22⋊C4—24no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42—81no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—48no—
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—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23C4×D412yes—
280C23C22×D411yescharacteristic
281D4—22no—
282D4—22no—
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—48no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—84no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4C4×D412yes—
330C2×C4C23×C411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C8—22no—
337C8—22no—
338C8—41no—
339C8—41no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—816no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C2×C4×D411yescharacteristic
370C22#53411yescharacteristic
371C22#53511yescharacteristic
372C22#57611yescharacteristic
373C22#57611yescharacteristic
374C22#57611yescharacteristic
375C22#57711yescharacteristic
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4#51811yescharacteristic
387C4#51811yescharacteristic
388C4#53811yescharacteristic
389C4#53811yescharacteristic
390C2—42no—
391C2—82no—
392C2—82no—
393C2—82no—
394C2#232111yescharacteristic
395C2#294611yescharacteristic
396C2#294811yescharacteristic
397C2#319811yescharacteristic
398C2#319911yescharacteristic
399C2#322011yescharacteristic
400C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2324C2semidirect
3. #2309C2semidirect
4. #1972C2semidirect
5. #1970C2semidirect
6. #1936C2semidirect
7. #1934C2semidirect
8. #1926C2semidirect
9. #1925C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1875C2semidirect
13. #1854C2semidirect
14. #1786C2semidirect
15. #1750C2semidirect
20. C2×C4×D4C22semidirect
23. #447C4semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
32. C2×#102C22semidirect
33. C2×#102C22semidirect
36. C2×#97C22semidirect
37. C2×C4×C8C22semidirect
45. #390C22semidirect
46. #389C22semidirect
53. C4×C4⋊C4C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #336C22semidirect
59. #336C22semidirect
68. C22×D4C2×C4semidirect
128. #102C2×C4semidirect
129. #102C2×C4semidirect
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×#4472
#23242
#23092
#19722
#19702
#19362
#19342
#19262
#19252
#18792
#18752
#18542
#17862
#17502