Finite Group Databasebeta

← #21457

Subgroups — #21457

Order 256.

734 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41C211yesmaximal, characteristic
2C2×#452C211yesmaximal, characteristic
3C2×#441C211yesmaximal, characteristic, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5C4×C22⋊Q8C211yesmaximal, characteristic, complemented
6#1878C211yesmaximal, characteristic, complemented
7#1877C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1749C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C4×#41—24no—
21C4×#41—24no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30C2×#117C2211yescharacteristic
31C22×C42—21no—
32C22×C42C2211yescharacteristic
33#452C418yescomplemented
34#441—24no—
35#440—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×#97C2211yescharacteristic
50#391C2211yescharacteristic
51#389C2211yescharacteristic
52#387—21no—
53#385—21no—
54#380C2211yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4C2211yescharacteristic
57C4×C22⋊C4—21no—
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#336C2211yescharacteristic
63#336C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41C2×C414yes—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4D411yescharacteristic
78C23×C4D411yescharacteristic
79C23×C4C2311yescharacteristic
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16C2×C414yescomplemented
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16C2×C414yescomplemented
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8C2×C414yescomplemented
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8D414yes—
98C4×D4—28no—
99C4×D4—28no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4D414yes—
104#117—28no—
105#117—22no—
106#117—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C42—28no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42D411yescharacteristic
131C2×C42D411yescharacteristic
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103C2×C414yescomplemented
141#102—22no—
142#102—22no—
143#102C2×C414yescomplemented
144C22⋊C8—22no—
145C22⋊C8C2×C414yescomplemented
146#97—22no—
147#97—22no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—21no—
154C24C2×D411yescharacteristic
155#41—28no—
156#41—28no—
157#41—28no—
158#41—28no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8C22×C412yes—
168C2×Q8C22×C412yes—
169C2×Q8C2×D412yes—
170C2×Q8#4112yes—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4C22×C412yes—
178C2×D4C2×D412yes—
179C2×D4#4112yes—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4C22×C412yes—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4#4112yes—
213C22×C4C2411yescharacteristic
214Q16—28no—
215SD16—28no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8C22×C412yes—
227C2×C8C22×C412yes—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C22×C412yes—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—42no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—24no—
252C42—22no—
253C42—21no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42C2×D412yes—
258C42C2×D412yes—
259C23—22no—
260C23—21no—
261C23—22no—
262C23—21no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C4×D412yes—
273C23C22×D411yescharacteristic
274Q8—22no—
275Q8—22no—
276Q8—22no—
277Q8—22no—
278Q8—22no—
279Q8—41no—
280Q8—41no—
281Q8C4×D414yes—
282D4—216no—
283D4—22no—
284D4—22no—
285D4—22no—
286D4C4×D414yes—
287C2×C4—216no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
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—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
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—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4C4×D412yes—
329C2×C4#12012yes—
330C2×C4#12012yes—
331C2×C4#13712yes—
332C2×C4C23×C411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C8—22no—
339C8—22no—
340C8—41no—
341C8—41no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—24no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—42no—
356C22C2×C4×D411yescharacteristic
357C22C2×#12011yescharacteristic
358C22#52511yescharacteristic
359C22C2×#13711yescharacteristic
360C22#57511yescharacteristic
361C22#57511yescharacteristic
362C22#57511yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—81no—
374C4—81no—
375C4C2×C4×D411yescharacteristic
376C4C2×C4×D411yescharacteristic
377C4#52511yescharacteristic
378C4#52511yescharacteristic
379C2—42no—
380C2—44no—
381C2C4×#12011yescharacteristic
382C2#294311yescharacteristic
383C2C4×#13711yescharacteristic
384C2#300511yescharacteristic
385C2#300711yescharacteristic
386C2#311111yescharacteristic
387C2#311611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#441C2semidirect
4. C2×#440C2semidirect
5. C4×C22⋊Q8C2semidirect
6. #1878C2semidirect
7. #1877C2semidirect
8. #1873C2semidirect
9. #1864C2semidirect
10. #1863C2semidirect
11. #1854C2semidirect
13. #1763C2semidirect
14. #1762C2semidirect
15. #1749C2semidirect
33. #452C4semidirect
82. C2×Q16C2×C4semidirect
85. C2×SD16C2×C4semidirect
91. C22⋊Q8C2×C4semidirect
140. #103C2×C4semidirect
143. #102C2×C4semidirect
145. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4522
C2×#4412
C2×#4402
C4×C22⋊Q82
#18782
#18772
#18732
#18642
#18632
#18542
#17882
#17632
#17622
#17492