Finite Group Databasebeta

← #10697586

Subgroups — #10697586

Order 768.

554 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3037C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#82479C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic, complemented
6#70507C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10244—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#464—22no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5229—21no—
19#5093—21no—
20#5076—21no—
21#5040—22no—
22#5019—22no—
23#4986—22no—
24#4977—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4929—22no—
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30#4535—22no—
31#4481—22no—
32#3206—31no—
33#3037D311yescharacteristic
34#2568—31no—
35#1548—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×Dic6D412yes—
42C3×#137—42no—
43C3×#137—42no—
44C2×C3×Q16—22no—
45C2×C3×SD16—22no—
46C2×C3×M16—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#214—24no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4×Dic3—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—22no—
74C3×#102—22no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#810—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×#137—21no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89#492—61no—
90#487—61no—
91#464—22no—
92#462—21no—
93#461—62no—
94#460—21no—
95#448—62no—
96#431—31no—
97#431—31no—
98#423C2×D311yescharacteristic
99#417—21no—
100#400—62no—
101#391—62no—
102#391—61no—
103#382—61no—
104C4×C4⋊C4—31no—
105#333—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—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8SD1612yescomplemented
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—22no—
130C22×Dic3—22no—
131C2×Dic6—22no—
132C2×Dic6—22no—
133C2×Dic6—44no—
134C3×Q16—44no—
135C3×SD16—44no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#214—84no—
154C4⋊Dic3—42no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C22×Q8—32no—
165C22×Q8#6711yescharacteristic
166#137—42no—
167#137—42no—
168C2×Q16—22no—
169C2×SD16—22no—
170C2×SD16—64no—
171C2×M16—21no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8#6711yescharacteristic
177#127#6711yescharacteristic
178#124—21no—
179#121—21no—
180#121—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—21no—
186C4×D4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8#6712yes—
199#103—22no—
200#103—64no—
201#102—22no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C4×C8—61no—
205C4×C8—61no—
206#95—32no—
207#95—64no—
208#95—64no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—22no—
212C3×Q8—22no—
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—41no—
216C3×Q8—41no—
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—42no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×C3×C4C2×SD1611yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4#13711yescharacteristic
243C2×Dic3—42no—
244C2×Dic3—44no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248Dic6—44no—
249Dic6—82no—
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—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—62no—
264C2×Q8—62no—
265C2×Q8—124no—
266C2×Q8#21412yescomplemented
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—44no—
280SD16—44no—
281SD16—244no—
282M16—41no—
283M16—41no—
284C2×C8—22no—
285C2×C8—62no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—122no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
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—42no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4#45011yescharacteristic
323C3×C4#45211yescharacteristic
324C3×C4#45512yes—
325C3×C4#46511yescharacteristic
326C3×C4#46811yescharacteristic
327Dic3—82no—
328Dic3—82no—
329C23—41no—
330C23#93711yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—124no—
338Q8—242no—
339D4—42no—
340D4—41no—
341D4—41no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—122no—
356C2×C4—124no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4#93211yescharacteristic
361C2×C4C2×#21411yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93811yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#94011yescharacteristic
367C2×C4#95011yescharacteristic
368C2×C4#95111yescharacteristic
369C2×C4#95311yescharacteristic
370C8—42no—
371C8—122no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#200011yescharacteristic
377C2×C3#200411yescharacteristic
378C2×C3#200911yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#503711yescharacteristic
385C22#504011yescharacteristic
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—242no—
394C4—242no—
395C4#504111yescharacteristic
396C4#504211yescharacteristic
397C4#520212yes—
398C4#521811yescharacteristic
399C4#523911yescharacteristic
400C3#1024411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2#7168511yescharacteristic
405C2#7210611yescharacteristic
406C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10244
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
3. #82479C2semidirect
5. #71621C2semidirect
6. #70507C2semidirect
7. #67715C2semidirect
118. C2×C3×Q8SD16semidirect
266. C2×Q8#214semidirect
400. C3#10244semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#833172
#824792
#718112
#716212
#705072
#677152
#102443