Finite Group Databasebeta

← #22477

Subgroups — #22477

Order 256.

647 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#551C211yesmaximal, characteristic
2#2342C212yesmaximal, complemented
3#1996C212yesmaximal, complemented
4#1899C212yesmaximal, complemented
5#1898C212yesmaximal
6#1820C211yesmaximal, characteristic, complemented
7#1819C211yesmaximal, characteristic, complemented
8#1799C211yesmaximal, characteristic, complemented
9C4×#99C211yesmaximal, characteristic, complemented
10#1740C211yesmaximal, characteristic, complemented
11C2×#342C211yesmaximal, characteristic, complemented
12#551—22no—
13#551—22no—
14#551—22no—
15#551—22no—
16C2×#126C411yescharacteristic, complemented
17C2×#126C411yescharacteristic, complemented
18C2×#124C411yescharacteristic, complemented
19C2×#123C411yescharacteristic, complemented
20C2×#123C411yescharacteristic, complemented
21C2×#123C2211yescharacteristic
22C2×C22⋊Q8C411yescharacteristic, complemented
23C2×C22⋊Q8C2211yescharacteristic
24C2×#120C2211yescharacteristic
25C2×C4×D4C411yescharacteristic, complemented
26C2×C4×D4C2211yescharacteristic
27C2×#117C411yescharacteristic, complemented
28C2×#117C2212yes—
29C2×#117C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2212yes—
32C22×C22⋊C4C2211yescharacteristic
33C2×#99C2212yes—
34C2×#99C2212yes—
35#388—22no—
36#388—22no—
37#385—22no—
38#381C2212yes—
39#380C2212yes—
40#379C2212yes—
41C4×C22⋊C4—22no—
42C4×C22⋊C4C2212yes—
43#342C2211yescharacteristic
44#342C2211yescharacteristic
45#342C2211yescharacteristic
46#342C2212yes—
47#342C2212yes—
48#342C2212yes—
49#342C2212yes—
50#342C2211yescharacteristic
51C25C2311yescharacteristic
52C22×Q8C2×C411yescharacteristic, complemented
53C22×D4C2×C411yescharacteristic
54C22×D4C2×C411yescharacteristic
55C22×D4C2311yescharacteristic
56C23×C4—22no—
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic, complemented
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62#126—22no—
63#126—22no—
64#126—22no—
65#126—22no—
66#124—22no—
67#124—22no—
68#123—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73#123—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78#120—22no—
79#120—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86#117—22no—
87#117—22no—
88#117—21no—
89#117—21no—
90#117—21no—
91#117—21no—
92#117—24no—
93#117—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4C2×C411yescharacteristic, complemented
97C2×C4⋊C4C2×C411yescharacteristic, complemented
98C2×C4⋊C4C2×C411yescharacteristic, complemented
99C2×C4⋊C4C2×C411yescharacteristic, complemented
100C2×C4⋊C4C2×C411yescharacteristic, complemented
101C2×C4⋊C4C2×C411yescharacteristic, complemented
102C2×C4⋊C4C2×C411yescharacteristic, complemented
103C2×C4⋊C4C2×C411yescharacteristic, complemented
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C22⋊C4C2×C411yescharacteristic
117C2×C22⋊C4C2×C411yescharacteristic
118C2×C22⋊C4C2×C411yescharacteristic
119C2×C22⋊C4C2×C411yescharacteristic, complemented
120C2×C22⋊C4C2×C411yescharacteristic, complemented
121C2×C22⋊C4C2×C411yescharacteristic, complemented
122C2×C22⋊C4D412yes—
123C2×C22⋊C4D412yes—
124C2×C22⋊C4D412yes—
125C2×C22⋊C4D412yes—
126C2×C22⋊C4C2312yes—
127C2×C22⋊C4C2312yes—
128C2×C22⋊C4C2312yes—
129C2×C22⋊C4C2311yescharacteristic
130C2×C22⋊C4C2311yescharacteristic
131C2×C22⋊C4C2311yescharacteristic
132C2×C22⋊C4C2312yes—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42C2×C411yescharacteristic, complemented
136C2×C42C2×C411yescharacteristic, complemented
137C2×C42C2×C411yescharacteristic, complemented
138#99—24no—
139#99—42no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144C24—22no—
145C24—21no—
146C24—21no—
147C24—42no—
148C24—81no—
149C24C22×C411yescharacteristic
150C24C22×C411yescharacteristic
151C24C22×C411yescharacteristic
152C24C2×D412yes—
153C24C2×D412yes—
154C24C2411yescharacteristic
155C2×Q8—22no—
156C2×Q8—81no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—81no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4C22×C411yescharacteristic
187C22×C4C22×C411yescharacteristic
188C22×C4C22×C411yescharacteristic
189C22×C4C22×C411yescharacteristic
190C22×C4C22×C411yescharacteristic
191C22×C4C22×C411yescharacteristic
192C22×C4C22×C411yescharacteristic
193C22×C4C22×C411yescharacteristic
194C22×C4C22×C411yescharacteristic
195C22×C4C22×C411yescharacteristic
196C22×C4C22×C411yescharacteristic
197C22×C4C2×D412yes—
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—82no—
236C22⋊C4C22⋊C411yescharacteristic
237C22⋊C4C22⋊C411yescharacteristic
238C22⋊C4C22⋊C411yescharacteristic
239C22⋊C4C22⋊C411yescharacteristic
240C22⋊C4C22⋊C412yes—
241C22⋊C4C22⋊C412yes—
242C22⋊C4C22⋊C412yes—
243C22⋊C4C22⋊C412yes—
244C22⋊C4C22⋊C412yes—
245C22⋊C4C22⋊C412yes—
246C22⋊C4#4114yes—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—44no—
254C42—82no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—42no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—84no—
279C23—82no—
280C23—82no—
281C23C2×C22⋊C412yes—
282C23C2×C22⋊C411yescharacteristic
283C23C2×C22⋊C411yescharacteristic
284C23C23×C411yescharacteristic
285C23C22×D412yes—
286C23C2×#4112yes—
287C23H(5,𝔽2)11yescharacteristic
288C232−1+411yescharacteristic
289Q8—82no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
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—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—88no—
341C2×C4C2×C22⋊C411yescharacteristic
342C2×C4C2×C22⋊C411yescharacteristic
343C2×C4C2×C22⋊C411yescharacteristic
344C2×C4C2×C22⋊C411yescharacteristic
345C2×C4C2×C22⋊C412yes—
346C2×C4C2×C22⋊C412yes—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—44no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—82no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#41012yes—
376C22C22×C22⋊C411yescharacteristic
377C22C4×#4111yescharacteristic
378C22#52011yescharacteristic
379C22#54612yes—
380C22#54712yes—
381C4—21no—
382C4—22no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—44no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4—82no—
398C2—22no—
399C2—22no—
400C2—22no—
401C2—21no—
402C2—21no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2#233911yescharacteristic
407C2C2×#41011yescharacteristic
408C2#288311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
prime radicalC2×#120C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2342C2semidirect
3. #1996C2semidirect
4. #1899C2semidirect
6. #1820C2semidirect
7. #1819C2semidirect
8. #1799C2semidirect
9. C4×#99C2semidirect
10. #1740C2semidirect
11. C2×#342C2semidirect
16. C2×#126C4semidirect
17. C2×#126C4semidirect
18. C2×#124C4semidirect
19. C2×#123C4semidirect
20. C2×#123C4semidirect
22. C2×C22⋊Q8C4semidirect
25. C2×C4×D4C4semidirect
27. C2×#117C4semidirect
52. C22×Q8C2×C4semidirect
59. C23×C4C2×C4semidirect
96. C2×C4⋊C4C2×C4semidirect
97. C2×C4⋊C4C2×C4semidirect
98. C2×C4⋊C4C2×C4semidirect
99. C2×C4⋊C4C2×C4semidirect
100. C2×C4⋊C4C2×C4semidirect
101. C2×C4⋊C4C2×C4semidirect
102. C2×C4⋊C4C2×C4semidirect
103. C2×C4⋊C4C2×C4semidirect
119. C2×C22⋊C4C2×C4semidirect
120. C2×C22⋊C4C2×C4semidirect
121. C2×C22⋊C4C2×C4semidirect
135. C2×C42C2×C4semidirect
136. C2×C42C2×C4semidirect
137. C2×C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5512
#23422
#19962
#18992
#18982
#18202
#18192
#17992
C4×#992
#17402
C2×#3422