Finite Group Databasebeta

← #10695934

Subgroups — #10695934

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#79309C211yesmaximal, characteristic, complemented
2#79055C211yesmaximal, characteristic
3#74190C211yesmaximal, characteristic, complemented
4#70757C211yesmaximal, characteristic, complemented
5#70500C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic, complemented
7C3×#1286C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C2×#908—21no—
10#5477C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5133—21no—
17#4988—21no—
18#4984—21no—
19#4925C2211yescharacteristic
20#4851—21no—
21#4848—21no—
22#4844—22no—
23#4843—21no—
24#4841—21no—
25#4738—21no—
26#4735—21no—
27C4×#204—22no—
28#4677—21no—
29#4673—21no—
30#4668—22no—
31#4484—22no—
32#4459—22no—
33#4451—22no—
34#3103—31no—
35#3036—31no—
36#2580—31no—
37#1517—31no—
38#1487—31no—
39#1478—31no—
40#1286D311yescharacteristic, complemented
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58#908—42no—
59#908—42no—
60C2×D24—22no—
61C2×#204—22no—
62C2×#204—22no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
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×C4⋊C8D412yes—
79C3×C22⋊C8—21no—
80C3×C4×C8D412yes—
81C3×#95—21no—
82C3×#95D412yes—
83C3⋊#95—22no—
84#830—21no—
85#821—22no—
86#821—22no—
87#817—22no—
88#816—22no—
89#816—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8—21no—
92C2×AGL(1,ℤ/8)—61no—
93#532—31no—
94C2×C4×Q8—31no—
95C2×C4×D4—31no—
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#431—31no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107#417—61no—
108#399—61no—
109#393—61no—
110#390—61no—
111#389—61no—
112#385—62no—
113#382C2×D311yescharacteristic
114#327—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C23×D3—21no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12SD1612yescomplemented
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—82no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C3×M16—41no—
148C3×M16—41no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—82no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—82no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174C2×C3⋊C8—41no—
175D24—44no—
176#204—44no—
177#204—44no—
178C2×#41—61no—
179C22×Q8—31no—
180C22×D4—31no—
181C23×C4—61no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—21no—
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—61no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—61no—
217C4⋊C8D1212yes—
218#103—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C4×C8—61no—
225C4×C8D1212yes—
226#95—21no—
227#95—62no—
228#95D1212yes—
229C23×C3#12011yescharacteristic
230C22×D3—22no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247C2×C3×C4C2×SD1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4#13711yescharacteristic
251#67—81no—
252#67—81no—
253C2×Dic3—22no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262D12—22no—
263D12—22no—
264D12—42no—
265D12—41no—
266D12—41no—
267D12—41no—
268D12—41no—
269D12—81no—
270C4×D3—82no—
271C4×D3—82no—
272C4×D3—81no—
273C4×D3—81no—
274Dic6—22no—
275Dic6—22no—
276Dic6—41no—
277Dic6—41no—
278Dic6—41no—
279Dic6—41no—
280C3×C8—42no—
281C3×C8—42no—
282C3⋊C8—81no—
283C24—61no—
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—32no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—121no—
302C22×C4—22no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—242no—
310C22×C4C2×D1211yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313SD16—124no—
314SD16—124no—
315D8—124no—
316M16—41no—
317M16—41no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—121no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—242no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—242no—
338C42—121no—
339C42—121no—
340C42C2×D1211yescharacteristic
341C42C2×D1211yescharacteristic
342C42D3×D411yescharacteristic
343C42#23911yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3#39211yescharacteristic
347C22×C3#44811yescharacteristic
348C22×C3#44911yescharacteristic
349C2×D3—44no—
350C2×D3—42no—
351C2×D3—82no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—82no—
362C3×C4#45011yescharacteristic
363C3×C4#45211yescharacteristic
364C3×C4#46511yescharacteristic
365C3×C4#46811yescharacteristic
366Dic3—42no—
367Dic3—81no—
368Dic3—81no—
369Dic3—81no—
370C23—62no—
371C23—122no—
372C23—122no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23#88211yescharacteristic
377Q8—62no—
378Q8—62no—
379Q8—121no—
380Q8—121no—
381Q8—121no—
382Q8—121no—
383D4—62no—
384D4—62no—
385D4—122no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—121no—
390D4—241no—
391D4—241no—
392D4—241no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—62no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—122no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—242no—
408C2×C4—242no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4#87411yescharacteristic
412C2×C4#88211yescharacteristic
413C2×C4#88211yescharacteristic
414C2×C4#89511yescharacteristic
415C2×C4#89511yescharacteristic
416C2×C4#89511yescharacteristic
417C2×C4#91111yescharacteristic
418C2×C4D3×SD1611yescharacteristic
419C2×C4#91611yescharacteristic
420C2×C4#91811yescharacteristic
421C8—42no—
422C8—42no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3#200011yescharacteristic
427C2×C3#200811yescharacteristic
428C2×C3#201011yescharacteristic
429D3—82no—
430D3—161no—
431C22—21no—
432C22—21no—
433C22—124no—
434C22—122no—
435C22—242no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22—241no—
440C22—241no—
441C22#467111yescharacteristic
442C22#478111yescharacteristic
443C22#480611yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—82no—
449C4—122no—
450C4—241no—
451C4—241no—
452C4—241no—
453C4#477611yescharacteristic
454C4#480811yescharacteristic
455C4#486411yescharacteristic
456C4#488611yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—242no—
461C2—481no—
462C2#6870311yescharacteristic
463C2#6888711yescharacteristic
464C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#392
prime radical#79055C2
hypercenter#1286D3
hypocenterC3#9998
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
1. #79309C2semidirect
3. #74190C2semidirect
4. #70757C2semidirect
5. #70500C2semidirect
6. #69545C2semidirect
7. C3×#1286C2semidirect
40. #1286D3semidirect
138. 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
#793092
#790552
#741902
#707572
#705002
#695452
C3×#12862
#99983