Finite Group Databasebeta

← #10695928

Subgroups — #10695928

Order 768.

602 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3103C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic, complemented
5#71622C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#949—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#481—21no—
14C3×#475—21no—
15C3×#460—21no—
16C3×#448—21no—
17C3×C4×SD16—22no—
18C3×#431C2211yescharacteristic
19#5232—21no—
20#5231—21no—
21#5083—21no—
22#5072—21no—
23#5057—22no—
24#4993—21no—
25#4989—22no—
26#4939—21no—
27#4936C2211yescharacteristic
28#4928—21no—
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4537—22no—
32#4535—22no—
33#4459—22no—
34#3103D311yescharacteristic, complemented
35#3036—31no—
36#2580—31no—
37#1517—31no—
38#1487—31no—
39#1478—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×SD16—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61#949—42no—
62#949—42no—
63C2×#215—22no—
64C2×#213—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—22no—
71C2×#208—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C3×#106—22no—
77C3×C4⋊C8—21no—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C4×C8D412yes—
82#835—22no—
83#832—22no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yes—
91C4×C3⋊C8—21no—
92C2×AGL(1,ℤ/8)—61no—
93#532C2×D311yescharacteristic
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×D4—31no—
96#481—21no—
97#475—21no—
98#463—62no—
99#461—61no—
100#460—21no—
101#459—61no—
102#448—21no—
103C4×SD16—22no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423—31no—
107#417—61no—
108#399—61no—
109#393—61no—
110#390—61no—
111#389—61no—
112#385—62no—
113#382—31no—
114#327—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C23×D3—21no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—42no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12SD1612yescomplemented
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—82no—
147C3×SD16—44no—
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#215—44no—
164#213—44no—
165(C2×D3)⋊C4—44no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—82no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—41no—
172C4⋊Dic3—41no—
173Dic3⋊C4—82no—
174#208—41no—
175#208—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×#41—21no—
179C22×Q8#6711yescharacteristic
180C22×D4—31no—
181C23×C4—61no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—22no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—61no—
188C4⋊Q8#6711yescharacteristic
189#127#6711yescharacteristic
190#124—21no—
191#121—21no—
192#121—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213#106—22no—
214C4⋊C8—21no—
215C4⋊C8—32no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218#103—22no—
219#103—62no—
220#102—22no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C4×C8—61no—
225C4×C8#6712yes—
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3—41no—
230C23×C3#12011yescharacteristic
231C22×D3—22no—
232C22×D3—42no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C3×Q8—22no—
237C3×Q8—22no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×D4—42no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—81no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×C3×C4C2×SD1611yescharacteristic
266C2×C3×C4#13511yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4#13711yescharacteristic
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—42no—
274D12—22no—
275D12—22no—
276D12—41no—
277D12—41no—
278C4×D3—82no—
279C4×D3—82no—
280C3×C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—61no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—32no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—242no—
310C22×C4D3×D411yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4C2×#6711yescharacteristic
313SD16—44no—
314SD16—124no—
315D8—124no—
316M16—121no—
317M16—121no—
318C2×C8—22no—
319C2×C8—62no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—242no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—124no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—242no—
338C42—41no—
339C42—41no—
340C42D3×D411yescharacteristic
341C42#23711yescharacteristic
342C42C2×#6711yescharacteristic
343C42C2×#6711yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#44811yescharacteristic
351C22×C3#44911yescharacteristic
352C2×D3—44no—
353C2×D3—42no—
354C2×D3—82no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—42no—
362C3×C4—81no—
363C3×C4—81no—
364C3×C4#45011yescharacteristic
365C3×C4#45211yescharacteristic
366C3×C4#46511yescharacteristic
367C3×C4#46811yescharacteristic
368Dic3—82no—
369Dic3—81no—
370C23—41no—
371C23—62no—
372C23—122no—
373C23—122no—
374C23—121no—
375C23—121no—
376C23#93711yescharacteristic
377Q8—22no—
378Q8—22no—
379Q8—41no—
380Q8—41no—
381Q8—41no—
382Q8—41no—
383D4—42no—
384D4—41no—
385D4—41no—
386D4—62no—
387D4—62no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—121no—
392D4—121no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—242no—
410C2×C4—242no—
411C2×C4D3×SD1611yescharacteristic
412C2×C4#91411yescharacteristic
413C2×C4#91511yescharacteristic
414C2×C4#91611yescharacteristic
415C2×C4#93711yescharacteristic
416C2×C4#93811yescharacteristic
417C2×C4#93911yescharacteristic
418C2×C4#93911yescharacteristic
419C2×C4#94011yescharacteristic
420C2×C4#95311yescharacteristic
421C8—42no—
422C8—122no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—161no—
427C2×C3#200011yescharacteristic
428C2×C3#200811yescharacteristic
429C2×C3#201011yescharacteristic
430D3—82no—
431C22—21no—
432C22—21no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—124no—
437C22—122no—
438C22—242no—
439C22—241no—
440C22—241no—
441C22#517011yescharacteristic
442C22#517211yescharacteristic
443C22#522311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—242no—
452C4—241no—
453C4#516911yescharacteristic
454C4#517111yescharacteristic
455C4#517311yescharacteristic
456C4#517411yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—161no—
461C2—242no—
462C2#7171511yescharacteristic
463C2#7205411yescharacteristic
464C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3103C2
socleC22×C3#392
prime radical#83297C2
hypercenter#3103D3
hypocenterC3#9998
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. C3×#3103C2semidirect
3. #82501C2semidirect
4. #71814C2semidirect
5. #71622C2semidirect
6. #70757C2semidirect
7. #67738C2semidirect
34. #3103D3semidirect
143. C2×D12SD16semidirect
457. C3#9998semidirect

Maximal subgroups

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

subgroupindex
C3×#31032
#832972
#825012
#718142
#716222
#707572
#677382
#99983