Finite Group Databasebeta

← #24395

Subgroups — #24395

Order 256.

725 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#408C211yesmaximal, characteristic, complemented
6#2616C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22#537—24no—
23#537—24no—
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121—21no—
31C2×#121—21no—
32C2×#121—21no—
33C2×#121C2211yescharacteristic
34C2×#121C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic
36#470—24no—
37#465—24no—
38#452—24no—
39C2×#106C2211yescharacteristic
40C2×#106C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49#408—24no—
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54#395—21no—
55#394—21no—
56#394—21no—
57#394—21no—
58#380C2211yescharacteristic
59#380C2211yescharacteristic
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64#340C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41—41no—
76C2×#41—41no—
77C22×Q8D411yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4D411yescharacteristic
84C22×D4D411yescharacteristic
85C22×D4D411yescharacteristic
86C22×D4C2311yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—41no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—41no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22⋊Q8—28no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C22⋊Q8—41no—
117C22⋊Q8—41no—
118C22⋊Q8D414yes—
119#121—28no—
120#121—28no—
121#121—28no—
122#121—22no—
123#121—22no—
124#121—22no—
125#121D414yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149#106—22no—
150#106—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—41no—
156#102—22no—
157#102—22no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8D414yescomplemented
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24C2×D411yescharacteristic
168C24C2×D411yescharacteristic
169C24C2×D411yescharacteristic
170#41—44no—
171#41—44no—
172#41—44no—
173#41—44no—
174#41—44no—
175#41—44no—
176#41—44no—
177#41—44no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8C2×D412yes—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—81no—
213C2×D4—81no—
214C2×D4—81no—
215C2×D4—81no—
216C2×D4C2×D412yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D412yes—
247C22×C4C2×D412yes—
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D412yes—
250C22×C4#4112yes—
251C22×C4C2411yescharacteristic
252Q16—44no—
253SD16—44no—
254M16—44no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—44no—
266C2×C8C2×D412yes—
267C2×C8C2×D412yes—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4C2×D412yes—
283C4⋊C4C2×D412yes—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23#12012yes—
321C23#12112yes—
322C23C22×D411yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—44no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332D4—22no—
333D4—22no—
334D4—44no—
335D4—44no—
336D4—44no—
337D4—48no—
338D4—41no—
339D4—41no—
340D4—84no—
341D4—84no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—48no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4#12012yes—
385C2×C4#12112yes—
386C2×C4C22×D411yescharacteristic
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C2×#4111yescharacteristic
390C2×C4H(5,𝔽2)11yescharacteristic
391C2×C4H(5,𝔽2)11yescharacteristic
392C8—22no—
393C8—22no—
394C8—41no—
395C8—41no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C2×#12011yescharacteristic
425C22C2×#12111yescharacteristic
426C22#53411yescharacteristic
427C22#57711yescharacteristic
428C22#57711yescharacteristic
429C22#57711yescharacteristic
430C22#57811yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4D4211yescharacteristic
442C4D4211yescharacteristic
443C4#54611yescharacteristic
444C4#54611yescharacteristic
445C2—42no—
446C2—42no—
447C2—81no—
448C2—82no—
449C2#241011yescharacteristic
450C2#300911yescharacteristic
451C2#301311yescharacteristic
452C2#306011yescharacteristic
453C2#306311yescharacteristic
454C2#320111yescharacteristic
455C2#320411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#470C2semidirect
3. C2×#465C2semidirect
4. C2×#452C2semidirect
5. C2×#408C2semidirect
6. #2616C2semidirect
7. #2035C2semidirect
8. #2034C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1895C2semidirect
12. #1892C2semidirect
13. #1872C2semidirect
14. #1853C2semidirect
15. #1788C2semidirect
162. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4702
C2×#4652
C2×#4522
C2×#4082
#26162
#20352
#20342
#20012
#20002
#18952
#18922
#18722
#18532
#17882