Finite Group Databasebeta

← #25534

Subgroups — #25534

Order 256.

606 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#466C211yesmaximal, characteristic, complemented
4#2414C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#553—24no—
19#553—24no—
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#482—24no—
35#466—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#390—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#126—28no—
85#125—22no—
86#125—22no—
87#123—28no—
88#121—28no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—41no—
96#121D414yescomplemented
97C4×D4—28no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102#117—22no—
103#117—22no—
104#117—41no—
105#117D414yescomplemented
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#107—22no—
136#106—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24—21no—
150C24C2×D411yescharacteristic
151C24C2×D411yescharacteristic
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—24no—
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—21no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4C2×D412yes—
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—24no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D412yes—
206C22×C4C2×D412yes—
207C22×C4C2411yescharacteristic
208D8—44no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—44no—
221C2×C8#4112yes—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—81no—
235C4⋊C4C2×D412yes—
236C4⋊C4C2×D412yes—
237C4⋊C4C2×D412yes—
238C4⋊C4#4112yes—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—22no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—24no—
252C42—21no—
253C42—21no—
254C42—42no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#12012yes—
282C23#13512yes—
283C23C22×D411yescharacteristic
284D4—22no—
285D4—22no—
286D4—44no—
287D4—48no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—84no—
295D4—84no—
296D4—84no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4#12012yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C42−1+411yescharacteristic
343C8—22no—
344C8—41no—
345C8—41no—
346C8—41no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22C2×#12011yescharacteristic
376C22#54611yescharacteristic
377C22#54711yescharacteristic
378C22C2×#13511yescharacteristic
379C22#57511yescharacteristic
380C22#57611yescharacteristic
381C22#57611yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—41no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4#53211yescharacteristic
393C4#53711yescharacteristic
394C4#54611yescharacteristic
395C4#54711yescharacteristic
396C2—42no—
397C2—42no—
398C2—82no—
399C2—161no—
400C2#243011yescharacteristic
401C2#300211yescharacteristic
402C2#300811yescharacteristic
403C2#329711yescharacteristic
404C2#330911yescharacteristic
405C2#333911yescharacteristic
406C2#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×#553C2semidirect
2. C2×#482C2semidirect
3. C2×#466C2semidirect
4. #2414C2semidirect
5. #2059C2semidirect
6. #2034C2semidirect
7. #2025C2semidirect
8. #1999C2semidirect
9. #1879C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1869C2semidirect
13. #1862C2semidirect
14. #1852C2semidirect
15. #1735C2semidirect
96. #121D4semidirect
105. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4822
C2×#4662
#24142
#20592
#20342
#20252
#19992
#18792
#18752
#18742
#18692
#18622
#18522
#17352