Finite Group Databasebeta

← #24762

Subgroups — #24762

Order 256.

530 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4C2×#424C211yesmaximal, characteristic, complemented
5#2619C211yesmaximal, characteristic, complemented
6#2324C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2054C211yesmaximal, characteristic, complemented
10#2054C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#469—24no—
27#466—24no—
28#459—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31#424—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44#394—21no—
45#394—21no—
46#392—21no—
47#392—21no—
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#386—21no—
55#385—21no—
56#382C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C2×D8—41no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—41no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#127—22no—
90#127—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121D414yescomplemented
97C4×D4—22no—
98C4×D4—22no—
99#117—22no—
100#117—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#107—22no—
126#106—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8D414yescomplemented
131#102—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—41no—
138#102—41no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C2×D4—24no—
149C2×D4—21no—
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—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4C2×D412yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201D8—44no—
202D8—44no—
203M16—44no—
204C2×C8—21no—
205C2×C8—21no—
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—44no—
215C2×C8C2×D412yes—
216C2×C8C2×D412yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
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—21no—
228C4⋊C4—81no—
229C4⋊C4C2×D412yes—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—81no—
246C42C2×D412yes—
247C42#4112yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
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—41no—
274C23—41no—
275C23#12112yes—
276C23C22×D411yescharacteristic
277D4—22no—
278D4—22no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—84no—
290D4—84no—
291D4—84no—
292D4—84no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
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—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4#12112yes—
327C2×C4#12112yes—
328C2×C4#12112yes—
329C2×C4C22×D411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C42−1+411yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#12111yescharacteristic
368C22C2×#12111yescharacteristic
369C22#53711yescharacteristic
370C22#57611yescharacteristic
371C22#57611yescharacteristic
372C22#57611yescharacteristic
373C22#57711yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4D4211yescharacteristic
385C4#54711yescharacteristic
386C4#54811yescharacteristic
387C4#55311yescharacteristic
388C2—42no—
389C2—82no—
390C2—161no—
391C2—161no—
392C2#241611yescharacteristic
393C2#304111yescharacteristic
394C2#304511yescharacteristic
395C2#305811yescharacteristic
396C2#306211yescharacteristic
397C2#325711yescharacteristic
398C2#325911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#469C2semidirect
2. C2×#466C2semidirect
3. C2×#459C2semidirect
4. C2×#424C2semidirect
5. #2619C2semidirect
6. #2324C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #2054C2semidirect
10. #2054C2semidirect
11. #1982C2semidirect
12. #1949C2semidirect
13. #1895C2semidirect
14. #1893C2semidirect
15. #1804C2semidirect
96. #121D4semidirect
130. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4662
C2×#4592
C2×#4242
#26192
#23242
#20672
#20652
#20542
#19822
#19492
#18952
#18932
#18042