Finite Group Databasebeta

← #10848604

Subgroups — #10848604

Order 768.

1188 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic, complemented
3#83184C211yesmaximal, characteristic, complemented
4#82533C211yesmaximal, characteristic, complemented
5#71857C214yesmaximal, complemented
6#71839C211yesmaximal, characteristic, complemented
7#71766C211yesmaximal, characteristic, complemented
8#71649C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25590—31nomaximal, Sylow
11C3×#559—22no—
12C3×#550—22no—
13C3×#546—24no—
14C3×#540—22no—
15C3×#535C2212yes—
16C3×#533C2211yescharacteristic
17C3×#532—22no—
18C2×C3×#127C2211yescharacteristic
19C3×#518C2212yes—
20C3×C4×#41C2211yescharacteristic
21C2×#951C2212yes—
22C2×#936C2214yes—
23C2×#869C2211yescharacteristic
24C2×#837C2212yes—
25#5221—24no—
26C2×#833C2214yes—
27#5093—22no—
28#5069—22no—
29#5031—22no—
30#5019—22no—
31#5004C2212yes—
32#5004C2214yes—
33C42×Dic3C2211yescharacteristic
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4532C2214yes—
37#4532C2212yes—
38#3508D311yescharacteristic
39#3139—31no—
40#3039—31no—
41#2858—31no—
42#2004—34no—
43#1994—31no—
44#1965—31no—
45#1885—34no—
46C4×#104—31no—
47C2×C3×#41—24no—
48C2×C3×#41C2311yescharacteristic
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C22×Dic6C2312yes—
52C3×C4⋊Q8D412yes—
53C3×#127—22no—
54C3×#127—22no—
55C3×#127—21no—
56C3×#127—21no—
57C3×#125D412yes—
58C3×#124—24no—
59C3×#124—22no—
60C3×#123—28no—
61C3×C22⋊Q8—22no—
62C3×#121—24no—
63C3×#121—28no—
64C3×#121—24no—
65C3×#120—24no—
66C3×C4×Q8—22no—
67C3×C4×Q8D412yes—
68C3×C4×D4—24no—
69C3×C4×D4—22no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4D412yes—
73C3×#117—21no—
74C3×#117C2311yescharacteristic
75C3×#117C2312yes—
76C2×C3×C22⋊C4—24no—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—28no—
80#951—44no—
81C2×#216—22no—
82#936—28no—
83#936—44no—
84C2×#214—22no—
85C2×C4⋊Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3C2312yes—
88C2×#208C2312yes—
89C2×#208C2312yes—
90#869—22no—
91#869—21no—
92#869—21no—
93#837—24no—
94#833—44no—
95#810—24no—
96#810—28no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137—32no—
101#559—22no—
102#550—22no—
103#546—24no—
104#540—22no—
105#535C2×D312yes—
106#533C2×D311yescharacteristic
107#532—22no—
108C2×C4⋊Q8—31no—
109C2×#127C2×D311yescharacteristic
110C2×#124—34no—
111#518C2×D312yes—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104—32no—
119#419—34no—
120#419—32no—
121C2×#100—34no—
122C4×M16—31no—
123#390—64no—
124C43—31no—
125#339—32no—
126#339—34no—
127C24×C3C2×D414yes—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—88no—
131C2×C3×Q8—22no—
132C2×C3×Q8C2×D411yescharacteristic
133C2×C3×D4—21no—
134C2×C3×D4—28no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—48no—
142C2×C3×D4C2×D411yescharacteristic
143C22×C3×C4—28no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—22no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—24no—
153C2×Dic6—24no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4C2×D412yes—
158C3×C4⋊C4C2×D414yes—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—28no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C42—22no—
165C3×C42—21no—
166C3×C42—22no—
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D412yes—
170#217—416no—
171#216—44no—
172#214—44no—
173C4⋊Dic3—42no—
174C4⋊Dic3—42no—
175C4×Dic3—22no—
176C4×Dic3—44no—
177C4×Dic3#4114yescomplemented
178#208—22no—
179#208—22no—
180#208—24no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×#41—24no—
184C2×#41C22×D311yescharacteristic
185C22×Q8—32no—
186C22×D4#6714yes—
187C22×D4C22×D312yes—
188#137—124no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×M16—32no—
192C2×M16—32no—
193C4⋊Q8—62no—
194C4⋊Q8—61no—
195C4⋊Q8—61no—
196C4⋊Q8#6712yes—
197#127—22no—
198#127—21no—
199#127—21no—
200#127—22no—
201#125#6712yes—
202#124—24no—
203#124—22no—
204#124—68no—
205#124—124no—
206#123—28no—
207C22⋊Q8—22no—
208#121—24no—
209#121—28no—
210#121—24no—
211#120—24no—
212C4×Q8—22no—
213C4×Q8#6712yes—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4#6712yes—
219#117—21no—
220#117C22×D311yescharacteristic
221#117C22×D312yes—
222C2×C4⋊C4—62no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—68no—
225C2×C42—21no—
226C2×C42—32no—
227C2×C42—62no—
228C2×C42C22×D311yescharacteristic
229C4⋊C8—62no—
230#104—64no—
231#103—64no—
232#103—68no—
233#100—124no—
234#97—61no—
235C4×C8—61no—
236C23×C3—24no—
237C23×C3—21no—
238C23×C3—28no—
239C23×C3—48no—
240C23×C3C22×D411yescharacteristic
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—44no—
244C3×D4—21no—
245C3×D4—21no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—44no—
250C3×D4—48no—
251C3×D4—84no—
252C3×D4—88no—
253C2×C3×C4—24no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—42no—
258C2×C3×C4—44no—
259C2×C3×C4—88no—
260C2×C3×C4#12012yes—
261C2×C3×C4#12012yes—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4112yes—
265C2×C3×C4H(5,𝔽2)12yes—
266C2×Dic3—24no—
267C2×Dic3—44no—
268C2×Dic3—42no—
269C2×Dic3—44no—
270Dic6—42no—
271Dic6—42no—
272Dic6—44no—
273C3⋊C8—42no—
274C3⋊C8—42no—
275C24C2×#6714yes—
276#41—22no—
277#41—22no—
278#41—88no—
279C2×Q8—22no—
280C2×Q8—64no—
281C2×Q8—64no—
282C2×Q8—62no—
283C2×Q8—62no—
284C2×Q8C2×#6711yescharacteristic
285C2×D4—21no—
286C2×D4—28no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—48no—
294C2×D4C2×#6711yescharacteristic
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4C2×#6712yes—
301C22×C4C23×D311yescharacteristic
302Q16—124no—
303SD16—124no—
304M16—62no—
305M16—62no—
306M16—64no—
307C2×C8—62no—
308C2×C8—62no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—122no—
313C4⋊C4—122no—
314C4⋊C4C2×#6712yes—
315C4⋊C4C2×#6714yes—
316C22⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—1216no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—34no—
326C42—62no—
327C42—124no—
328C42C2×#6711yescharacteristic
329C42C2×#6711yescharacteristic
330C42C2×#6712yes—
331C22×C3—22no—
332C22×C3—41no—
333C22×C3—44no—
334C22×C3—42no—
335C22×C3—48no—
336C22×C3—88no—
337C22×C3C2×#12011yescharacteristic
338C22×C3#54612yes—
339C3×C4—22no—
340C3×C4—41no—
341C3×C4—42no—
342C3×C4—44no—
343C3×C4#45514yes—
344C3×C4#53211yescharacteristic
345C3×C4#53511yescharacteristic
346C3×C4#55012yes—
347Dic3—44no—
348Dic3—82no—
349C23—24no—
350C23—21no—
351C23—28no—
352C23—48no—
353C23C22×#6711yescharacteristic
354Q8—21no—
355Q8—21no—
356Q8—44no—
357Q8—122no—
358Q8—122no—
359Q8—124no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—48no—
364D4—42no—
365D4—42no—
366D4—44no—
367D4—84no—
368D4—88no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—64no—
376C2×C4—88no—
377C2×C4—124no—
378C2×C4—124no—
379C2×C4—122no—
380C2×C4#95312yes—
381C2×C4#95312yes—
382C2×C4D3×#4112yes—
383C2×C4#100912yes—
384C2×C4C22×#6711yescharacteristic
385C2×C4C22×#6711yescharacteristic
386C8—122no—
387C8—122no—
388C2×C3—22no—
389C2×C3—81no—
390C2×C3—84no—
391C2×C3#243111yescharacteristic
392C2×C3C2×#45512yes—
393C22—22no—
394C22—41no—
395C22—44no—
396C22—42no—
397C22—48no—
398C22—88no—
399C22#582912yes—
400C22C2×#95311yescharacteristic
401C4—22no—
402C4—41no—
403C4—42no—
404C4—44no—
405C4—124no—
406C4—242no—
407C4#524214yes—
408C4#583011yescharacteristic
409C4#583211yescharacteristic
410C4#583412yes—
411C3#2559011yescharacteristic, complemented, Sylow
412C2—22no—
413C2—84no—
414C2—81no—
415C2#8409211yescharacteristic
416C2C2×#524212yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3508C2
socleC22×C3C2×#120
hypercenter#3508D3
hypocenterC3#25590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
2. #83337C2semidirect
3. #83184C2semidirect
4. #82533C2semidirect
5. #71857C2semidirect
6. #71839C2semidirect
7. #71766C2semidirect
8. #71649C2semidirect
9. C4×#837C2semidirect
177. C4×Dic3#41semidirect
411. C3#25590semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#833372
#831842
#825332
#718572
#718392
#717662
#716492
C4×#8372
#255903