Finite Group Databasebeta

← #10695936

Subgroups — #10695936

Order 768.

554 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 101 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#83315C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic, complemented
6#70500C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9999—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#464—22no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5229—21no—
19#5094—21no—
20#5075—21no—
21#5040—21no—
22#5034—21no—
23C4×#214—22no—
24#4986—21no—
25#4979—21no—
26#4972—21no—
27#4936C2211yescharacteristic
28#4933—22no—
29#4925C2211yescharacteristic
30#4925C2211yescharacteristic
31#4537—22no—
32#4535—22no—
33#4484—22no—
34#3103—31no—
35#3037D311yescharacteristic
36#2581—31no—
37#1517—31no—
38#1487—31no—
39#1479—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C3×#137—42no—
45C3×#137—42no—
46C2×C3×Q16—22no—
47C2×C3×SD16—22no—
48C2×C3×M16—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#214—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×Dic3⋊C4—22no—
68C2×Dic3⋊C4—21no—
69C2×C4×Dic3—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×#103—22no—
77C3×#102—22no—
78#832—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#810—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92C2×#137—21no—
93#532C2×D311yescharacteristic
94C2×C4×Q8—31no—
95C2×C4×Q8C2×D311yescharacteristic
96#481—61no—
97#475—61no—
98#464—22no—
99#462—21no—
100#460—21no—
101#460—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#423C2×D311yescharacteristic
107#417—21no—
108#400—61no—
109#395—61no—
110#391—61no—
111#389—61no—
112#387—62no—
113#382—31no—
114#327—62no—
115Q8⋊C8—62no—
116Q8⋊C8—62no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8SD1612yescomplemented
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—41no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3—21no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6—42no—
144C3×Q16—44no—
145C3×SD16—44no—
146C3×M16—41no—
147C3×M16—41no—
148C2×C3×C8—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#214—44no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168Dic3⋊C4—44no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—41no—
172C4×Dic3—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×#41—21no—
176C22×Q8—31no—
177C22×Q8#6711yescharacteristic
178#137—42no—
179#137—42no—
180C2×Q16—22no—
181C2×SD16—22no—
182C2×SD16—62no—
183C2×M16—21no—
184C4⋊Q8#6711yescharacteristic
185#127#6711yescharacteristic
186#124—21no—
187#121—21no—
188#121—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—21no—
198C4×D4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C42—21no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#106—62no—
209C4⋊C8—21no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8#6712yes—
213#103—22no—
214#103—62no—
215#102—22no—
216#102—62no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C4×C8—32no—
220C4×C8—61no—
221#95—32no—
222#95—62no—
223#95—62no—
224#95—62no—
225#95—61no—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C3×Q8—22no—
229C3×Q8—22no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×D4—42no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—81no—
238C3×D4—81no—
239C3×D4—81no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256C2×C3×C4C2×SD1611yescharacteristic
257C2×C3×C4#13511yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4#13711yescharacteristic
260C2×Dic3—22no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268Dic6—22no—
269Dic6—22no—
270Dic6—41no—
271Dic6—41no—
272C3×C8—42no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—122no—
288C2×Q8#21412yescomplemented
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—61no—
297C22×C4—62no—
298C22×C4—61no—
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—44no—
303SD16—44no—
304SD16—124no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—62no—
309C2×C8—121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—124no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C4⋊C4—121no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42—121no—
328C42—121no—
329C42D3×D411yescharacteristic
330C42#23711yescharacteristic
331C42C2×#6711yescharacteristic
332C42C2×#6711yescharacteristic
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3#39211yescharacteristic
339C22×C3#44811yescharacteristic
340C22×C3#44911yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—81no—
347C3×C4—81no—
348C3×C4#45011yescharacteristic
349C3×C4#45211yescharacteristic
350C3×C4#46511yescharacteristic
351C3×C4#46811yescharacteristic
352Dic3—42no—
353Dic3—82no—
354Dic3—81no—
355C23—41no—
356C23#93711yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—62no—
364Q8—62no—
365Q8—121no—
366Q8—121no—
367D4—42no—
368D4—41no—
369D4—41no—
370D4—81no—
371D4—81no—
372D4—81no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—62no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4#93211yescharacteristic
392C2×C4C2×#21411yescharacteristic
393C2×C4#93711yescharacteristic
394C2×C4#93811yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#94011yescharacteristic
398C2×C4#95011yescharacteristic
399C2×C4#95111yescharacteristic
400C2×C4#95311yescharacteristic
401C8—42no—
402C8—122no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—161no—
407C2×C3#200011yescharacteristic
408C2×C3#200911yescharacteristic
409C2×C3#201011yescharacteristic
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22#503711yescharacteristic
416C22#504011yescharacteristic
417C22#522311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—81no—
425C4—122no—
426C4—242no—
427C4—241no—
428C4#504111yescharacteristic
429C4#504211yescharacteristic
430C4#521811yescharacteristic
431C4#523911yescharacteristic
432C3#999911yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—161no—
436C2#7168511yescharacteristic
437C2#7211111yescharacteristic
438C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#9999
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. #82481C2semidirect
5. #71621C2semidirect
6. #70500C2semidirect
7. #67713C2semidirect
127. C2×C3×Q8SD16semidirect
288. C2×Q8#214semidirect
432. C3#9999semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#833152
#824812
#718142
#716212
#705002
#677132
#99993