Finite Group Databasebeta

← #22158

Subgroups — #22158

Order 256.

464 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic
2#2328C211yesmaximal, characteristic, complemented
3#2319C211yesmaximal, characteristic, complemented
4#1982C211yesmaximal, characteristic, complemented
5#1972C211yesmaximal, characteristic, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1949C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1768C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#459C418yescomplemented
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#386—21no—
46#386—21no—
47#386—21no—
48#386—21no—
49#382C2211yescharacteristic
50#381—21no—
51#380—21no—
52C4×C4⋊C4C2211yescharacteristic
53#376C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8—41no—
71C2×D8—41no—
72C2×D8C2×C414yescomplemented
73C2×D8C2×C414yescomplemented
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#127—22no—
79#127C2×C414yes—
80C4×D4—22no—
81C4×D4—41no—
82C4×D4—41no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4C2×C414yescomplemented
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—41no—
109#106—41no—
110C4⋊C8—22no—
111C4⋊C8C2×C414yescomplemented
112#102—41no—
113#102—41no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102C2×C414yescomplemented
119#102C2×C414yescomplemented
120#97—41no—
121C4×C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95D411yescharacteristic
128#95D411yescharacteristic
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C24—21no—
133C24—21no—
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—81no—
152C2×D4—81no—
153C2×D4—81no—
154C2×D4—81no—
155C2×D4C22×C412yes—
156C2×D4C22×C412yes—
157C2×D4C22×C412yes—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2411yescharacteristic
179D8—44no—
180D8—44no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8C22×C412yes—
192C2×C8C22×C412yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4C22×C412yes—
205C22⋊C4—24no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—81no—
221C42C22×C412yes—
222C42#4112yes—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23C22×D411yescharacteristic
245D4—22no—
246D4—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—84no—
258D4—84no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
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—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4C4×D412yes—
294C2×C4C4×D412yes—
295C2×C4C23×C411yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C42−1+411yescharacteristic
301C8—22no—
302C8—22no—
303C8—41no—
304C8—41no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×C4×D411yescharacteristic
327C22#53511yescharacteristic
328C22#53711yescharacteristic
329C22#57611yescharacteristic
330C22#57611yescharacteristic
331C22#57611yescharacteristic
332C22#57711yescharacteristic
333C4—22no—
334C4—22no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4#51811yescharacteristic
345C4#52011yescharacteristic
346C4#54011yescharacteristic
347C4#54211yescharacteristic
348C2—82no—
349C2—82no—
350C2—82no—
351C2#232411yescharacteristic
352C2#294611yescharacteristic
353C2#294811yescharacteristic
354C2#322011yescharacteristic
355C2#322411yescharacteristic
356C2#325711yescharacteristic
357C2#325911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2328C2semidirect
3. #2319C2semidirect
4. #1982C2semidirect
5. #1972C2semidirect
6. #1970C2semidirect
7. #1949C2semidirect
8. #1936C2semidirect
9. #1934C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1864C2semidirect
13. #1862C2semidirect
14. #1804C2semidirect
15. #1768C2semidirect
22. #459C4semidirect
72. C2×D8C2×C4semidirect
73. C2×D8C2×C4semidirect
85. C4×D4C2×C4semidirect
111. C4⋊C8C2×C4semidirect
118. #102C2×C4semidirect
119. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
#23282
#23192
#19822
#19722
#19702
#19492
#19362
#19342
#18792
#18642
#18622
#18042
#17682