Finite Group Databasebeta

← #10695933

Subgroups — #10695933

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
1#82388C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#79309C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic, complemented
6#70770C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5221—21no—
16#5210—22no—
17#5019—21no—
18#5016—22no—
19#5015—21no—
20#4972—21no—
21#4966—21no—
22#4949—21no—
23#4947—21no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4851—21no—
27#4848—21no—
28#4738—21no—
29#4735—21no—
30C4×#204—22no—
31C3×D4⋊C8—22no—
32#4470—22no—
33#4456—22no—
34#3103—31no—
35#3036—31no—
36#2580—31no—
37#1517—31no—
38#1487—31no—
39#1478D311yescharacteristic, complemented
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×Dic6D411yescharacteristic
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—22no—
54C2×#214—22no—
55#932—42no—
56#932—42no—
57C2×#213—22no—
58#930—21no—
59#930—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—22no—
64C2×C4×Dic3—21no—
65C2×#208—21no—
66C2×#204—22no—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C4×Dic6—21no—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C4×Dic6—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C4×C8—21no—
81C3×C4×C8D412yes—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#821—22no—
86#817—22no—
87#816—22no—
88#810—22no—
89#809—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92C2×AGL(1,ℤ/8)—61no—
93#532—31no—
94C2×C4×Q8—31no—
95C2×C4×D4C2×D311yescharacteristic
96#481—61no—
97#475—61no—
98#463—62no—
99#461—61no—
100#460—61no—
101#459—61no—
102#448—61no—
103C4×SD16—62no—
104#431C2×D311yescharacteristic
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—22no—
117C24×C3—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4SD1612yescomplemented
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—82no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C2×#67—41no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135#235—41no—
136#235—41no—
137#235—41no—
138#235—41no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×C4×D3—41no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157#217—44no—
158#217—82no—
159#214—44no—
160#213—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—82no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174#208—41no—
175#208—41no—
176C2×C3⋊C8—22no—
177#204—44no—
178C2×#41—61no—
179C22×Q8—31no—
180C22×D4D1211yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—61no—
188C4⋊Q8—31no—
189#127—31no—
190#124—61no—
191#121—61no—
192#121—61no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4D1211yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—32no—
217C4⋊C8—61no—
218#103—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C4×C8—21no—
225C4×C8#6712yes—
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3#12011yescharacteristic
235C22×D3—41no—
236C3×D4—22no—
237C3×D4—22no—
238C3×D4—41no—
239C3×D4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—82no—
245C2×C3×C4—82no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×C3×C4C2×SD1611yescharacteristic
253C2×C3×C4#13511yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
255C2×C3×C4#13711yescharacteristic
256#67—81no—
257#67—81no—
258C2×Dic3—22no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268D12—42no—
269D12—41no—
270D12—41no—
271D12—81no—
272C4×D3—81no—
273C4×D3—81no—
274Dic6—22no—
275Dic6—22no—
276Dic6—41no—
277Dic6—41no—
278Dic6—41no—
279Dic6—41no—
280C3×C8—42no—
281C3×C8—81no—
282C3⋊C8—42no—
283C24—21no—
284#41—121no—
285#41—121no—
286#41—121no—
287#41—121no—
288C2×Q8—61no—
289C2×Q8—61no—
290C2×Q8—61no—
291C2×Q8—122no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—121no—
301C2×D4#20412yescomplemented
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—61no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—82no—
309C22×C4—121no—
310C22×C4C2×D1211yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313SD16—124no—
314SD16—124no—
315D8—124no—
316M16—121no—
317M16—121no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—121no—
328C4⋊C4—121no—
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—121no—
339C42—121no—
340C42#23511yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C42C2×#6711yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—44no—
347C22×C3—42no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#39211yescharacteristic
352C22×C3#44811yescharacteristic
353C22×C3#44911yescharacteristic
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—81no—
362C3×C4#45011yescharacteristic
363C3×C4#45211yescharacteristic
364C3×C4#46511yescharacteristic
365C3×C4#46811yescharacteristic
366Dic3—42no—
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—121no—
376C23#88211yescharacteristic
377Q8—62no—
378Q8—62no—
379Q8—121no—
380Q8—121no—
381Q8—121no—
382Q8—121no—
383D4—22no—
384D4—22no—
385D4—41no—
386D4—41no—
387D4—122no—
388D4—121no—
389D4—121no—
390D4—241no—
391D4—241no—
392D4—241no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—62no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4#88211yescharacteristic
412C2×C4#88411yescharacteristic
413C2×C4#88411yescharacteristic
414C2×C4C2×#20411yescharacteristic
415C2×C4#90411yescharacteristic
416C2×C4#90811yescharacteristic
417C2×C4#90911yescharacteristic
418C2×C4#93011yescharacteristic
419C2×C4#93711yescharacteristic
420C2×C4#94011yescharacteristic
421C8—42no—
422C8—81no—
423C8—122no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—82no—
427C2×C3#200011yescharacteristic
428C2×C3#200811yescharacteristic
429C2×C3#201011yescharacteristic
430D3—161no—
431C22—21no—
432C22—21no—
433C22—44no—
434C22—42no—
435C22—82no—
436C22—81no—
437C22—81no—
438C22—241no—
439C22—241no—
440C22—241no—
441C22#473411yescharacteristic
442C22#473811yescharacteristic
443C22#496011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—122no—
450C4—242no—
451C4—241no—
452C4—241no—
453C4#473111yescharacteristic
454C4#473311yescharacteristic
455C4#511411yescharacteristic
456C4#513411yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—82no—
461C2—481no—
462C2#6996211yescharacteristic
463C2#7033111yescharacteristic
464C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radical#82307C2
hypercenter#1478D3
hypocenterC3#9998
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #82388C2semidirect
3. #79309C2semidirect
4. C3×#1478C2semidirect
5. #70790C2semidirect
6. #70770C2semidirect
7. #67738C2semidirect
39. #1478D3semidirect
123. C2×C3×D4SD16semidirect
301. C2×D4#204semidirect
457. C3#9998semidirect

Maximal subgroups

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

subgroupindex
#823882
#823072
#793092
C3×#14782
#707902
#707702
#677382
#99983