Finite Group Databasebeta

← #30469

Subgroups — #30469

Order 256.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic
5#2434C211yesmaximal, characteristic, complemented
6#2427C211yesmaximal, characteristic
7#2056C211yesmaximal, characteristic, complemented
8#2043C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic
13#1935C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1877C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#497—24no—
31#461—24no—
32#451—24no—
33#448—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46#397—21no—
47#395—21no—
48#395—21no—
49#394—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#391C2211yescharacteristic
53#389C2211yescharacteristic
54#386—21no—
55#385—21no—
56#385—21no—
57#380—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—41no—
84C2×Q16—41no—
85C2×Q16—41no—
86C2×Q16—41no—
87C2×Q16—41no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16—22no—
91C2×SD16—22no—
92C2×SD16—41no—
93C2×SD16—41no—
94C2×SD16—41no—
95C2×SD16—41no—
96C2×SD16D414yes—
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105#124—22no—
106#124—41no—
107C22⋊Q8—22no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111#121—22no—
112#121—41no—
113C4×D4—22no—
114C4×D4—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4D411yescharacteristic
132C2×C22⋊C4D411yescharacteristic
133C2×C42C2311yescharacteristic
134C2×C42C2311yescharacteristic
135C4⋊C8—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141#103—41no—
142#103D414yescomplemented
143#102—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146#97—22no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95D411yescharacteristic
155#95D411yescharacteristic
156C24—21no—
157C24C2×D411yescharacteristic
158C2×Q8—21no—
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×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
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—21no—
179C2×Q8—21no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8C2×D412yes—
188C2×Q8C2×D412yes—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—81no—
197C2×D4C2×D412yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223Q16—44no—
224Q16—44no—
225Q16—44no—
226SD16—44no—
227SD16—44no—
228SD16—44no—
229C2×C8—21no—
230C2×C8—21no—
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×C8C2×D412yes—
240C2×C8C2×D412yes—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4C2×D412yes—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#13712yes—
280C23C22×D411yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—22no—
284Q8—22no—
285Q8—48no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298D4—22no—
299D4—22no—
300D4—41no—
301D4—41no—
302D4—84no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—48no—
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—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—84no—
344C2×C4#13712yes—
345C2×C4C22×D411yescharacteristic
346C2×C4C22×D411yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C8—22no—
352C8—22no—
353C8—41no—
354C8—41no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#53511yescharacteristic
370C22D4211yescharacteristic
371C22D4211yescharacteristic
372C22C2×#13711yescharacteristic
373C22C2×#13711yescharacteristic
374C22#57711yescharacteristic
375C22#57811yescharacteristic
376C4—22no—
377C4—41no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4#53411yescharacteristic
389C4#53511yescharacteristic
390C4D4211yescharacteristic
391C4#56011yescharacteristic
392C2—42no—
393C2—82no—
394C2#266911yescharacteristic
395C2#321811yescharacteristic
396C2#322511yescharacteristic
397C2#326611yescharacteristic
398C2#326611yescharacteristic
399C2#327011yescharacteristic
400C2#327611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#497C2semidirect
2. C2×#461C2semidirect
3. C2×#451C2semidirect
5. #2434C2semidirect
7. #2056C2semidirect
8. #2043C2semidirect
9. #2016C2semidirect
10. #2015C2semidirect
11. #2001C2semidirect
13. #1935C2semidirect
14. #1924C2semidirect
15. #1877C2semidirect
142. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4612
C2×#4512
C2×#4482
#24342
#24272
#20562
#20432
#20162
#20152
#20012
#20002
#19352
#19242
#18772