Finite Group Databasebeta

← #28237

Subgroups — #28237

Order 256.

548 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×#469C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic, complemented
4C2×#435C211yesmaximal, characteristic, complemented
5#2537C211yesmaximal, characteristic, complemented
6#2535C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12#1809C211yesmaximal, characteristic, complemented
13#1807C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#469—24no—
28#466—24no—
29#442—24no—
30#435—24no—
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46#397—21no—
47#394—21no—
48#394—21no—
49#392—21no—
50#386—21no—
51#386—21no—
52#384C2211yescharacteristic
53#382C2211yescharacteristic
54#380—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×D8—22no—
75C2×D8—22no—
76C2×D8D414yescomplemented
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#121—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—41no—
92#121—41no—
93#121—41no—
94#121—41no—
95#121—44no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—41no—
126#106—22no—
127#106—22no—
128#106—41no—
129C4⋊C8—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138#97D414yescomplemented
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C24C2×D411yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C2×D4C2×D412yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205D8—28no—
206M16—44no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—44no—
216C2×C8C2×D412yes—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C2×C8#4112yes—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42C2×D412yes—
248C42#4112yes—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23AGL(1,ℤ/8)12yes—
277C23C22×D411yescharacteristic
278D4—22no—
279D4—22no—
280D4—22no—
281D4—22no—
282D4—44no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—84no—
288D4—84no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
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—84no—
328C2×C4—84no—
329C2×C4#12112yes—
330C2×C4#12112yes—
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C2×C42−1+411yescharacteristic
337C8—22no—
338C8—41no—
339C8—41no—
340C8#12114yescomplemented
341C22—22no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C2×#12111yescharacteristic
370C22D4211yescharacteristic
371C22#54711yescharacteristic
372C22C2×AGL(1,ℤ/8)11yescharacteristic
373C22#57511yescharacteristic
374C22#57611yescharacteristic
375C22#57711yescharacteristic
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4C2×#12111yescharacteristic
387C4#53711yescharacteristic
388C4#54711yescharacteristic
389C4#55511yescharacteristic
390C2—42no—
391C2—81no—
392C2—82no—
393C2—82no—
394C2#253711yescharacteristic
395C2#305511yescharacteristic
396C2#306211yescharacteristic
397C2#326411yescharacteristic
398C2#326711yescharacteristic
399C2#334511yescharacteristic
400C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#469C2semidirect
2. C2×#466C2semidirect
3. C2×#442C2semidirect
4. C2×#435C2semidirect
5. #2537C2semidirect
6. #2535C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #2033C2semidirect
11. #1847C2semidirect
12. #1809C2semidirect
13. #1807C2semidirect
14. #1804C2semidirect
15. #1802C2semidirect
76. C2×D8D4semidirect
138. #97D4semidirect
340. C8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4662
C2×#4422
C2×#4352
#25372
#25352
#20342
#20332
#18472
#18092
#18072
#18042
#18022