Finite Group Databasebeta

← #10697594

Subgroups — #10697594

Order 768.

558 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3227C211yesmaximal, characteristic
2#83314C211yesmaximal, characteristic
3#83243C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic, complemented
6#70766C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10245—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#951—21no—
12C2×C4×Dic6C2211yescharacteristic
13C3×#498—21no—
14C3×#486—21no—
15C3×#481—22no—
16C3×#476—22no—
17C3×#431C2211yescharacteristic
18#5232—21no—
19#5230—22no—
20#5090—21no—
21#5075—21no—
22#5054—22no—
23#4975—21no—
24#4973—21no—
25#4972—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4537—22no—
31#4484—22no—
32#3227D311yescharacteristic
33#3037—31no—
34#2567—31no—
35#1549—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56#951—42no—
57#951—42no—
58C2×#216—22no—
59C2×#214—22no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—24no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#106—24no—
72C3×C4⋊C8D412yes—
73C3×#102—24no—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76#835—22no—
77#832—22no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80#830—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×Q8—31no—
89#498—21no—
90#486—21no—
91#481—22no—
92#476—22no—
93#464—62no—
94#462—61no—
95#460—61no—
96#431—31no—
97#431C2×D311yescharacteristic
98#423—31no—
99#417—61no—
100#395—62no—
101#391—61no—
102#389—62no—
103#384—61no—
104C4×C4⋊C4—31no—
105#332—62no—
106#327—62no—
107Q8⋊C8—62no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—82no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C22×C3×C4—22no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6SD1612yes—
134C2×C3×C8—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#216—44no—
148#214—44no—
149C4⋊Dic3—24no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—84no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C22×Q8—21no—
165C22×Q8—31no—
166#137—122no—
167#137—122no—
168C2×Q16—62no—
169C2×SD16—62no—
170C2×M16—61no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8#6711yescharacteristic
176#127#6711yescharacteristic
177#124—21no—
178#121—21no—
179#121—21no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—21no—
185C4×D4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4#6712yes—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—24no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—62no—
201#102—24no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C4×C8—21no—
206C4×C8—61no—
207#95—32no—
208#95—64no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—82no—
214C3×Q8—82no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—42no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—44no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234C2×C3×C4C2×SD1611yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×Dic3—24no—
239C2×Dic3—22no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246Dic6—22no—
247Dic6—22no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3⋊C8—42no—
252C3⋊C8—81no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—32no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—82no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—124no—
280SD16—124no—
281M16—121no—
282M16—121no—
283C2×C8—22no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—64no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—244no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—121no—
301C42—121no—
302C42—121no—
303C42—121no—
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C2×#6711yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C22×C3#44811yescharacteristic
315C22×C3#44911yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4—82no—
321C3×C4#45011yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#45612yes—
324C3×C4#46511yescharacteristic
325C3×C4#46811yescharacteristic
326Dic3—44no—
327Dic3—42no—
328Dic3—81no—
329C23—41no—
330C23#93711yescharacteristic
331Q8—41no—
332Q8—41no—
333Q8—62no—
334Q8—62no—
335Q8—82no—
336Q8—82no—
337Q8—121no—
338Q8—121no—
339D4—41no—
340D4—41no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—64no—
351C2×C4—62no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4D3×SD1611yescharacteristic
361C2×C4#91411yescharacteristic
362C2×C4#91511yescharacteristic
363C2×C4#91611yescharacteristic
364C2×C4#93711yescharacteristic
365C2×C4#93811yescharacteristic
366C2×C4#93911yescharacteristic
367C2×C4#93911yescharacteristic
368C2×C4#94011yescharacteristic
369C2×C4#95311yescharacteristic
370C8—42no—
371C8—122no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#200011yescharacteristic
377C2×C3#200511yescharacteristic
378C2×C3#200811yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#517011yescharacteristic
385C22#517211yescharacteristic
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—82no—
392C4—124no—
393C4—122no—
394C4—241no—
395C4#516911yescharacteristic
396C4#517111yescharacteristic
397C4#517311yescharacteristic
398C4#517411yescharacteristic
399C4#524612yes—
400C3#1024511yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2#7186811yescharacteristic
405C2#7205411yescharacteristic
406C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3227C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3227D3
hypocenterC3#10245
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
3. #83243C2semidirect
5. #71775C2semidirect
6. #70766C2semidirect
7. #67746C2semidirect
400. C3#10245semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#833142
#832432
#718112
#717752
#707662
#677462
#102453