Finite Group Databasebeta

← #29677

Subgroups — #29677

Order 256.

569 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2015C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic
13#1803C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20#542—24no—
21#542—24no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#126—21no—
24C2×#124—21no—
25C2×#124C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×#123—21no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×C4×Q8C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C2×#117C2211yescharacteristic
36#464—24no—
37#463—24no—
38C2×C4⋊C8C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50#396—21no—
51#396—21no—
52#394—21no—
53#394—21no—
54#391C2211yescharacteristic
55#390C2211yescharacteristic
56#384C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×Q16—22no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90#126—28no—
91#124—28no—
92#124—22no—
93#124—22no—
94#124—22no—
95#124—22no—
96#123—28no—
97C22⋊Q8—28no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102#121—22no—
103#121—22no—
104#121—41no—
105#121—41no—
106C4×Q8—22no—
107C4×Q8D414yescomplemented
108C4×D4—22no—
109C4×D4D414yescomplemented
110#117—22no—
111#117—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4D411yescharacteristic
119C2×C4⋊C4D411yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C22⋊C4—21no—
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⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42—21no—
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140C4⋊C8—22no—
141C4⋊C8—22no—
142#103—22no—
143#103—22no—
144#103—41no—
145#103—41no—
146#102—22no—
147#102—22no—
148#102—41no—
149#102—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157C2×Q8—24no—
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—81no—
170C2×Q8—81no—
171C2×Q8C2×D412yes—
172C2×D4—24no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—42no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4C2×D412yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—24no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211Q16—44no—
212SD16—44no—
213SD16—44no—
214D8—44no—
215C2×C8—21no—
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×C8—21no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—24no—
257C42—21no—
258C42—21no—
259C42C2×D412yes—
260C42C2×D412yes—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
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—41no—
282C23C22×D411yescharacteristic
283Q8—22no—
284Q8—22no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—84no—
292D4—22no—
293D4—22no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—84no—
301D4—84no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
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—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—84no—
338C2×C4#12012yes—
339C2×C4#12012yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C42−1+411yescharacteristic
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C2×#12011yescharacteristic
372C22#53511yescharacteristic
373C22#53711yescharacteristic
374C22#57611yescharacteristic
375C22#57711yescharacteristic
376C22#57711yescharacteristic
377C22#57811yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—42no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4#53511yescharacteristic
390C4#53511yescharacteristic
391C4#53711yescharacteristic
392C4#53711yescharacteristic
393C2—81no—
394C2—82no—
395C2—161no—
396C2#261511yescharacteristic
397C2#301211yescharacteristic
398C2#301311yescharacteristic
399C2#322411yescharacteristic
400C2#322511yescharacteristic
401C2#326211yescharacteristic
402C2#326311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. C2×#464C2semidirect
3. C2×#463C2semidirect
4. #2620C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #2015C2semidirect
8. #2012C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #2000C2semidirect
13. #1803C2semidirect
14. #1802C2semidirect
15. #1798C2semidirect
107. C4×Q8D4semidirect
109. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4642
C2×#4632
#26202
#20242
#20232
#20152
#20122
#20012
#20002
#19992
#18032
#18022
#17982