Finite Group Databasebeta

← #25397

Subgroups — #25397

Order 256.

559 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#435C211yesmaximal, characteristic, complemented
5#2540C211yesmaximal, characteristic, complemented
6#2537C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1947C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#468—24no—
31#466—24no—
32#463—24no—
33#435—24no—
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×#97C2211yescharacteristic
48#399—21no—
49#394—21no—
50#394—21no—
51#392—21no—
52#390—21no—
53#390—21no—
54#384C2211yescharacteristic
55#380—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#124—22no—
91#124—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104#121—44no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#107—22no—
136C4⋊C8D414yescomplemented
137#103—22no—
138#103—22no—
139#103—41no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—41no—
144#102D414yescomplemented
145C22⋊C8—22no—
146C22⋊C8—22no—
147#97—22no—
148#95—21no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
154C24C2×D411yescharacteristic
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4C2×D412yes—
188C22×C4—21no—
189C22×C4—21no—
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—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2411yescharacteristic
214SD16—44no—
215D8—44no—
216M16—44no—
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—44no—
227C2×C8C2×D412yes—
228C2×C8C2×D412yes—
229C2×C8C2×D412yes—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—81no—
243C4⋊C4C2×D412yes—
244C4⋊C4#4112yes—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—81no—
259C42C2×D412yes—
260C42#4112yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23AGL(1,ℤ/8)12yes—
289C23C22×D411yescharacteristic
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294D4—22no—
295D4—22no—
296D4—44no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—84no—
304D4—84no—
305D4—84no—
306D4—84no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
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—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—84no—
344C2×C4#12112yes—
345C2×C4#12112yes—
346C2×C4C22×D411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C8—22no—
353C8—22no—
354C8—22no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C2×#12111yescharacteristic
385C22D4211yescharacteristic
386C22#54611yescharacteristic
387C22C2×AGL(1,ℤ/8)11yescharacteristic
388C22#57511yescharacteristic
389C22#57611yescharacteristic
390C22#57811yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—42no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4#53211yescharacteristic
402C4#54011yescharacteristic
403C4D4211yescharacteristic
404C4#55011yescharacteristic
405C2—42no—
406C2—81no—
407C2—82no—
408C2—161no—
409C2#242811yescharacteristic
410C2#305511yescharacteristic
411C2#306111yescharacteristic
412C2#326411yescharacteristic
413C2#326911yescharacteristic
414C2#330811yescharacteristic
415C2#331911yescharacteristic
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×#468C2semidirect
2. C2×#466C2semidirect
3. C2×#463C2semidirect
4. C2×#435C2semidirect
5. #2540C2semidirect
6. #2537C2semidirect
7. #2036C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1959C2semidirect
12. #1947C2semidirect
13. #1895C2semidirect
14. #1892C2semidirect
15. #1762C2semidirect
136. C4⋊C8D4semidirect
144. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4662
C2×#4632
C2×#4352
#25402
#25372
#20362
#20332
#20002
#19992
#19592
#19472
#18952
#18922
#17622