Finite Group Databasebeta

← #21772

Subgroups — #21772

Order 256.

695 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#408C211yesmaximal, characteristic, complemented
3#2311C211yesmaximal, characteristic, complemented
4#1895C211yesmaximal, characteristic, complemented
5#1894C211yesmaximal, characteristic, complemented
6#1893C211yesmaximal, characteristic, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1824C211yesmaximal, characteristic, complemented
13#1823C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#537—24no—
21#537C418yescomplemented
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121—21no—
27C2×#121—21no—
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41#408—24no—
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#391—21no—
47#390—21no—
48#386—21no—
49#386—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—41no—
71C2×#41C2×C414yes—
72C2×#41C2×C414yescomplemented
73C2×#41D414yes—
74C22×Q8D411yescharacteristic
75C22×Q8C2311yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4D411yescharacteristic
80C22×D4D411yescharacteristic
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22⋊Q8—28no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8C2×C414yes—
103C22⋊Q8C2×C414yescomplemented
104#121—28no—
105#121—28no—
106#121—28no—
107#121—22no—
108#121—22no—
109#121C2×C414yes—
110#121C2×C414yescomplemented
111#117—41no—
112#117—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C2×C414yescomplemented
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
136C2×C42—21no—
137#103—41no—
138#103—41no—
139#103—41no—
140#103—41no—
141#102—41no—
142#102—41no—
143#102—41no—
144#102—41no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—21no—
154C24C2×D411yescharacteristic
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157#41—44no—
158#41—44no—
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
163#41—44no—
164#41—44no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8C22⋊C414yes—
173C2×Q8C22×C412yes—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4C22⋊C414yes—
195C2×D4C22⋊C414yes—
196C2×D4C22⋊C414yes—
197C2×D4C22×C412yes—
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—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4C22×C412yes—
227C22×C4C22×C412yes—
228C22×C4C22×C412yes—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D412yes—
233C22×C4C2×D412yes—
234C22×C4C2×D412yes—
235C22×C4C2411yescharacteristic
236M16—44no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—44no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—42no—
264C4⋊C4C22×C412yes—
265C4⋊C4C22×C412yes—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×C22⋊C412yes—
302C23C2×C22⋊C412yes—
303C23C2×C22⋊C412yes—
304C23C22×D411yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—44no—
308Q8—41no—
309Q8—41no—
310D4—22no—
311D4—22no—
312D4—44no—
313D4—44no—
314D4—44no—
315D4—48no—
316D4—41no—
317D4—41no—
318D4—84no—
319D4—84no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—48no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4C2×C22⋊C412yes—
363C2×C4C2×C22⋊C412yes—
364C2×C4C2×C22⋊C412yes—
365C2×C4C23×C411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C8—41no—
372C8—41no—
373C8—41no—
374C8—41no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22C22×C22⋊C411yescharacteristic
404C22#53411yescharacteristic
405C22#53411yescharacteristic
406C22#57611yescharacteristic
407C22#57711yescharacteristic
408C22#57711yescharacteristic
409C22#57811yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4#51811yescharacteristic
421C4#51811yescharacteristic
422C4#53411yescharacteristic
423C4#53411yescharacteristic
424C2—42no—
425C2—42no—
426C2—42no—
427C2—82no—
428C2#230811yescharacteristic
429C2#289711yescharacteristic
430C2#289811yescharacteristic
431C2#319811yescharacteristic
432C2#320111yescharacteristic
433C2#320311yescharacteristic
434C2#320411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C2×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×#408C2semidirect
3. #2311C2semidirect
4. #1895C2semidirect
5. #1894C2semidirect
6. #1893C2semidirect
7. #1892C2semidirect
8. #1875C2semidirect
9. #1874C2semidirect
10. #1873C2semidirect
11. #1872C2semidirect
12. #1824C2semidirect
13. #1823C2semidirect
14. #1788C2semidirect
15. #1788C2semidirect
21. #537C4semidirect
72. C2×#41C2×C4semidirect
103. C22⋊Q8C2×C4semidirect
110. #121C2×C4semidirect
120. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4082
#23112
#18952
#18942
#18932
#18922
#18752
#18742
#18732
#18722
#18242
#18232
#17882