Finite Group Databasebeta

← #22425

Subgroups — #22425

Order 256.

723 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic
3C2×#419C211yesmaximal, characteristic, complemented
4C22×#103C211yesmaximal, characteristic, complemented
5#2352C211yesmaximal, characteristic
6#1981C211yesmaximal, characteristic, complemented
7#1960C211yesmaximal, characteristic, complemented
8#1894C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13#1787C211yesmaximal, characteristic, complemented
14#1787C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20D4×Q8—24no—
21D4×Q8C418yescomplemented
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#435—24no—
34C2×C4⋊C8C2211yescharacteristic
35#419—24no—
36C2×#103—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#103C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×#97C2211yescharacteristic
52#388—21no—
53#387—21no—
54#386—21no—
55#382C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59C4×C22⋊C4—21no—
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64C22×Q8—21no—
65C22×Q8—21no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C414yes—
77C22×Q8C2×C414yescomplemented
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4D411yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—41no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C4⋊Q8—28no—
98C4⋊Q8—22no—
99C4⋊Q8C2×C414yescomplemented
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8—28no—
103C22⋊Q8—28no—
104C22⋊Q8C2×C414yes—
105C22⋊Q8C2×C414yescomplemented
106C4×Q8—22no—
107C4×Q8C2×C414yescomplemented
108C4×D4—28no—
109C4×D4C2×C414yes—
110C4×D4D414yes—
111#117—22no—
112#117—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4D411yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141#103—22no—
142#103—22no—
143#103—22no—
144#103—22no—
145#103—22no—
146#103—41no—
147#103—41no—
148#103—41no—
149#103—41no—
150#103D414yescomplemented
151C22⋊C8—22no—
152C22⋊C8—22no—
153#97—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C24C2×D411yescharacteristic
160C24C2×D411yescharacteristic
161C2×Q8—24no—
162C2×Q8—24no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—28no—
169C2×Q8—28no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8C22×C412yes—
189C2×Q8C22×C412yes—
190C2×Q8C22×C412yes—
191C2×Q8C2×D412yes—
192C2×Q8#4112yes—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4C22⋊C414yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4C22×C412yes—
222C22×C4C22×C412yes—
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D412yes—
228C22×C4C2×D412yes—
229C22×C4C2411yescharacteristic
230M16—44no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—44no—
243C2×C8C2×D412yes—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4C22⋊C414yes—
259C4⋊C4C22×C412yes—
260C4⋊C4C2×D412yes—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C22⋊C4C22⋊C414yes—
267C22⋊C4C22⋊C414yes—
268C42—24no—
269C42—42no—
270C42—81no—
271C42C22×C412yes—
272C42C2×D412yes—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23C2×C22⋊C412yes—
284C23C2×C22⋊C412yes—
285C23#13712yes—
286C23#13712yes—
287C23C22×D411yescharacteristic
288Q8—22no—
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—22no—
294Q8—22no—
295Q8—48no—
296Q8—48no—
297Q8—41no—
298Q8—41no—
299Q8—41no—
300Q8—41no—
301Q8C4×D414yes—
302D4—28no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4C2×C22⋊C412yes—
347C2×C4C2×C22⋊C412yes—
348C2×C4C2×C22⋊C412yes—
349C2×C4C2×C22⋊C412yes—
350C2×C4C23×C411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C2×C4C22×D411yescharacteristic
353C2×C4C2×#4111yescharacteristic
354C2×C4H(5,𝔽2)11yescharacteristic
355C2×C4H(5,𝔽2)11yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#41914yescomplemented
374C22C22×C22⋊C411yescharacteristic
375C22D4211yescharacteristic
376C22#54611yescharacteristic
377C22C2×#13711yescharacteristic
378C22C2×#13711yescharacteristic
379C22#57711yescharacteristic
380C22#57811yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—22no—
385C4—22no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4C2×C4×D411yescharacteristic
394C4#51811yescharacteristic
395C4D4211yescharacteristic
396C4#54611yescharacteristic
397C2—24no—
398C2—42no—
399C2D4×C22⋊C411yescharacteristic
400C2C2×#41911yescharacteristic
401C2#289811yescharacteristic
402C2#326611yescharacteristic
403C2#327011yescharacteristic
404C2#331211yescharacteristic
405C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×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
3. C2×#419C2semidirect
4. C22×#103C2semidirect
6. #1981C2semidirect
7. #1960C2semidirect
8. #1894C2semidirect
9. #1892C2semidirect
10. #1873C2semidirect
11. #1872C2semidirect
12. #1805C2semidirect
13. #1787C2semidirect
14. #1787C2semidirect
15. #1763C2semidirect
21. D4×Q8C4semidirect
77. C22×Q8C2×C4semidirect
99. C4⋊Q8C2×C4semidirect
105. C22⋊Q8C2×C4semidirect
107. C4×Q8C2×C4semidirect
150. #103D4semidirect
373. C22#419semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4352
C2×#4192
C22×#1032
#23522
#19812
#19602
#18942
#18922
#18732
#18722
#18052
#17872
#17632