Finite Group Databasebeta

← #10831729

Subgroups — #10831729

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic
4#74190C211yesmaximal, characteristic
5C3×#1805C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70240C212yesmaximal
8#70238C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68899C212yesmaximal
11#68828C212yesmaximal, complemented
12#24340—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C22×#215C2211yescharacteristic
16#5570—24no—
17#5570—24no—
18C2×#890—22no—
19C2×#890C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869—21no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#382C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2212yes—
30C2×#817C2212yes—
31#5026—24no—
32#4988C2212yes—
33#4984C2212yes—
34C2×#809C2211yescharacteristic
35C2×#807C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4842—24no—
38#4677—21no—
39#4673—21no—
40#4668—22no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#476—31no—
48C2×#440—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1880—32no—
55#1865—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6—21no—
61C22×Dic6D411yescharacteristic
62C22×C3×C8C2312yes—
63C3×C4×Q8—22no—
64C3×C4×Q8D414yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70Q8×Dic3—24no—
71Q8×Dic3—24no—
72C2×#215—22no—
73C2×#215—22no—
74C2×#215Q814yescomplemented
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4D412yes—
87C2×C4×Dic3—22no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#890—216no—
92#890—24no—
93#890—24no—
94C4×D12—22no—
95C4×D12—22no—
96#869—28no—
97#869—22no—
98#869D414yes—
99C4×Dic6—28no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×C4⋊C8—22no—
103C3×#103—24no—
104C3×#95—21no—
105C3×#95—21no—
106#835—22no—
107C3⋊#95—22no—
108#821—24no—
109#817—24no—
110#809—22no—
111#807—22no—
112#803—22no—
113C22×SD16—31no—
114#557—64no—
115#557—64no—
116C2×C4⋊Q8—31no—
117C2×C4⋊Q8—61no—
118C2×#125—32no—
119C2×#125—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125#476—64no—
126#440—64no—
127C2×#107—31no—
128C2×#106—32no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—31no—
131C2×#103C2×D312yes—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382C2×D311yescharacteristic
141#340—32no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8C2×D412yes—
151C2×C3×Q8C2×Q812yes—
152C2×C3×Q8#4112yes—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C2×Q812yes—
169C2×C4×D3—42no—
170C2×C4×D3—82no—
171C2×Dic6—22no—
172C2×Dic6—24no—
173C2×Dic6—42no—
174C2×Dic6—44no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4C2×D412yes—
183C3×C42—24no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186#215—28no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—82no—
190C4⋊Dic3—28no—
191C4⋊Dic3—28no—
192C4⋊Dic3—22no—
193C4⋊Dic3—24no—
194C4⋊Dic3—22no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—82no—
198C4⋊Dic3C2×D414yes—
199Dic3⋊C4—28no—
200Dic3⋊C4—24no—
201C4×Dic3—28no—
202C4×Dic3—22no—
203C4×Dic3—22no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C2×Q812yes—
210C22×Q8—31no—
211C22×Q8—61no—
212C22×Q8C22×D311yescharacteristic
213C22×D4—31no—
214C23×C4—61no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8C22×D312yes—
221C4⋊Q8—34no—
222C4⋊Q8—68no—
223C4⋊Q8—62no—
224#125—64no—
225#125—616no—
226#125—64no—
227C4×Q8—22no—
228C4×Q8—68no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8D1214yes—
234C4×D4—62no—
235C4×D4—62no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4C22×D312yes—
247C2×C4⋊C4C22×D311yescharacteristic
248C2×C22⋊C4—62no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—62no—
251C2×C42—21no—
252C2×C42—32no—
253C2×C42—62no—
254C2×C42C22×D311yescharacteristic
255#107—62no—
256#106—64no—
257C4⋊C8—22no—
258#103—24no—
259#103—62no—
260#102—64no—
261#102—62no—
262#97—62no—
263#95—21no—
264#95—21no—
265#95—62no—
266C23×C3C22×D411yescharacteristic
267C22×D3—22no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—42no—
271C22×D3—42no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—22no—
275C3×Q8C22⋊Q814yes—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#12012yes—
287C2×C3×C4#12012yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—44no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304D12—22no—
305D12—22no—
306D12—41no—
307D12—41no—
308C4×D3—84no—
309Dic6—44no—
310Dic6—48no—
311C3×C8—42no—
312C3⋊C8—22no—
313C3⋊C8—41no—
314C24—61no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—64no—
319C2×Q8—62no—
320C2×Q8—122no—
321C2×Q8—124no—
322C2×Q8C2×D1212yes—
323C2×Q8#23712yes—
324C2×Q8D3×Q812yes—
325C2×D4—32no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—61no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—122no—
339C22×C4—242no—
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C23×D311yescharacteristic
343SD16—68no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—32no—
348C2×C8—61no—
349C2×C8—61no—
350C2×C8—61no—
351C2×C8—61no—
352C2×C8—61no—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—34no—
358C4⋊C4—68no—
359C4⋊C4—68no—
360C4⋊C4—68no—
361C4⋊C4—62no—
362C4⋊C4—64no—
363C4⋊C4—64no—
364C4⋊C4—62no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—242no—
368C4⋊C4C2×D1212yes—
369C22⋊C4—124no—
370C22⋊C4—122no—
371C22⋊C4—242no—
372C42—24no—
373C42—68no—
374C42—62no—
375C42—62no—
376C42C2×D1212yes—
377C42D3×D412yes—
378C22×C3C2×#12011yescharacteristic
379C22×C3#54811yescharacteristic
380C22×C3D4×Q811yescharacteristic
381C22×C3#57512yes—
382C22×C3#57712yes—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—82no—
386C2×D3—81no—
387C2×D3—81no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—82no—
393C3×C4C2×C22⋊Q811yescharacteristic
394C3×C4#53611yescharacteristic
395C3×C4#54811yescharacteristic
396C3×C4D4×Q811yescharacteristic
397Dic3—42no—
398Dic3—44no—
399Dic3—42no—
400Dic3—81no—
401C23—62no—
402C23—121no—
403C23—122no—
404C23—122no—
405C23—121no—
406C23C2×D3×D411yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—22no—
410Q8—124no—
411Q8—128no—
412Q8#89714yes—
413D4—62no—
414D4—62no—
415D4—121no—
416D4—121no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—62no—
428C2×C4—64no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—124no—
435C2×C4—121no—
436C2×C4—122no—
437C2×C4—244no—
438C2×C4#88212yes—
439C2×C4#88212yes—
440C2×C4C22×D1211yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23711yescharacteristic
443C2×C4C2×D3×Q811yescharacteristic
444C2×C4#101011yescharacteristic
445C2×C4#101011yescharacteristic
446C8—42no—
447C8—62no—
448C8—121no—
449C2×C3#240911yescharacteristic
450C2×C3#300511yescharacteristic
451C2×C3#301011yescharacteristic
452C2×C3#337112yes—
453C2×C3#338812yes—
454D3—82no—
455C22—122no—
456C22—124no—
457C22—241no—
458C22—242no—
459C22—241no—
460C22C2×#88211yescharacteristic
461C22#559811yescharacteristic
462C22#563211yescharacteristic
463C22#577012yes—
464C22#577712yes—
465C4—24no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—82no—
470C4—122no—
471C4—124no—
472C4—122no—
473C4—241no—
474C4C2×#89711yescharacteristic
475C4#552611yescharacteristic
476C4#559811yescharacteristic
477C4#563211yescharacteristic
478C3#2434011yescharacteristic, complemented, Sylow
479C2—242no—
480C2#7401711yescharacteristic
481C2#7883611yescharacteristic
482C2#7895911yescharacteristic
483C2#8058412yes—
484C2#8122512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#24340
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#5570C2semidirect
5. C3×#1805C2semidirect
6. #70473C2semidirect
8. #70238C2semidirect
9. #69555C2semidirect
11. #68828C2semidirect
56. #1805D3semidirect
74. C2×#215Q8semidirect
478. C3#24340semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
C2×#50262
C2×#48422
#741902
C3×#18052
#704732
#702402
#702382
#695552
#688992
#688282
#243403