Finite Group Databasebeta

← #10848842

Subgroups — #10848842

Order 768.

1094 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×#5731C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic, complemented
3C2×#4697C211yesmaximal, characteristic, complemented
4#74370C211yesmaximal, characteristic
5C3×#1956C211yesmaximal, characteristic, complemented
6#70273C212yesmaximal, complemented
7#70272C212yesmaximal
8#70240C212yesmaximal, complemented
9#70239C212yesmaximal, complemented
10#69753C211yesmaximal, characteristic, complemented
11#69583C211yesmaximal, characteristic, complemented
12#25612—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×#216C2212yes—
17C22×#215C2212yes—
18#5731—24no—
19#5731—24no—
20C2×#898—22no—
21C2×#897C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×#889—22no—
24C2×#875C2211yescharacteristic
25C2×#872C2211yescharacteristic
26C2×#870C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#816C2212yes—
34#5086—24no—
35#4994—22no—
36#4991—22no—
37#4988C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#803C2211yescharacteristic
40#4697—24no—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C2×#541—31no—
45C2×#497—31no—
46C2×#488—31no—
47#2620—31no—
48#2001—32no—
49#2000—32no—
50#1992—31no—
51#1956D311yescharacteristic, complemented
52#1880—32no—
53#1878—32no—
54#1750—31no—
55C22×C3×Q8C2312yes—
56C22×D12—21no—
57C22×D12C2311yescharacteristic
58C22×C4×D3C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yescomplemented
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68#946—24no—
69#946—24no—
70Q8×Dic3—24no—
71Q8×Dic3—24no—
72C2×#216—24no—
73C2×#216—24no—
74C2×#216—42no—
75C2×#215—24no—
76C2×#215—24no—
77C2×#215—42no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4D412yes—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3—22no—
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#898—216no—
94#897—42no—
95#895—42no—
96#889—216no—
97#875—22no—
98#875—22no—
99#872—22no—
100#872—22no—
101#870—22no—
102#870D414yescomplemented
103C3×#103—42no—
104C3×#103—42no—
105C3×C4×C8—22no—
106C3⋊#95—22no—
107#821—24no—
108(C2×D3)⋊C8—24no—
109#816—24no—
110#803—22no—
111C22×Q16—32no—
112C22×SD16—32no—
113#541—64no—
114#541—64no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#126—62no—
117C2×#125—31no—
118C2×#125—62no—
119C2×#124—31no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—31no—
122C2×#121—31no—
123C2×C4×Q8—32no—
124C2×#117—31no—
125#497—64no—
126#488—64no—
127C2×#103—32no—
128C2×#103C2×D312yes—
129C2×#103C2×D312yes—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8C2×D311yescharacteristic
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384C2×D311yescharacteristic
139#340—32no—
140#340—32no—
141#336—32no—
142C23×D3—21no—
143C23×D3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—22no—
156C22×Dic3—21no—
157C22×Dic3C2×D412yes—
158C22×Dic3C2×D411yescharacteristic
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—42no—
164C2×D12—82no—
165C2×C4×D3—21no—
166C2×C4×D3—21no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—82no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182#216—48no—
183#215—48no—
184(C2×D3)⋊C4—28no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—82no—
190C4⋊Dic3—28no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3C2×D414yes—
194Dic3⋊C4—28no—
195Dic3⋊C4—28no—
196Dic3⋊C4—24no—
197C4×Dic3—28no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8—31no—
204C22×Q8C22×D312yes—
205C22×D4—31no—
206C22×D4—61no—
207C23×C4—31no—
208C2×Q16—64no—
209C2×Q16—64no—
210C2×Q16—122no—
211C2×SD16—64no—
212C2×SD16—64no—
213C2×SD16—122no—
214C22×C8—32no—
215C22×C8C22×D312yes—
216C4⋊Q8—22no—
217C4⋊Q8D1214yescomplemented
218#126—616no—
219#125—34no—
220#125—616no—
221#125—62no—
222#124—62no—
223#124—62no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—122no—
227#121—122no—
228C4×Q8—64no—
229C4×Q8—64no—
230#117—62no—
231#117—62no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C22⋊C4—32no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C42—31no—
249C2×C42—62no—
250C2×C42C22×D311yescharacteristic
251#103—42no—
252#103—42no—
253#103—64no—
254#102—64no—
255C22⋊C8—64no—
256#97—62no—
257C4×C8—22no—
258#95—62no—
259C23×C3C22×D411yescharacteristic
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12014yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4H(5,𝔽2)12yes—
288C2×Dic3—22no—
289C2×Dic3—24no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300D12—42no—
301D12—42no—
302D12—84no—
303C4×D3—42no—
304C4×D3—42no—
305Dic6—42no—
306Dic6—42no—
307C3×C8—42no—
308C3⋊C8—24no—
309C24—31no—
310C24—61no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8C2×D1214yes—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—122no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—32no—
331C22×C4—31no—
332C22×C4—62no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4D3×D412yes—
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—34no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—34no—
352C4⋊C4—68no—
353C4⋊C4—68no—
354C4⋊C4—64no—
355C4⋊C4—68no—
356C4⋊C4—62no—
357C4⋊C4—82no—
358C4⋊C4—122no—
359C22⋊C4—68no—
360C22⋊C4—64no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C22⋊C4—124no—
364C22⋊C4—242no—
365C42—68no—
366C42—61no—
367C42—61no—
368C42C2×D1212yes—
369C42D3×D412yes—
370C22×C3C2×#12011yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3#53511yescharacteristic
373C22×C3#57712yes—
374C22×C3#57812yes—
375C2×D3—42no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—42no—
380C2×D3—82no—
381C2×D3—81no—
382C2×D3—81no—
383C2×D3—81no—
384C2×D3—82no—
385C3×C4—24no—
386C3×C4—44no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#54612yes—
390C3×C4#55012yes—
391Dic3—41no—
392Dic3—42no—
393Dic3—44no—
394Dic3—81no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—62no—
399C23—62no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—122no—
404C23—122no—
405C23C2×D3×D411yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—122no—
411Q8—122no—
412D4—122no—
413D4—122no—
414D4—244no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—64no—
428C2×C4—61no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4#88214yes—
439C2×C4C22×D1211yescharacteristic
440C2×C4C2×D3×D411yescharacteristic
441C2×C4#100412yes—
442C2×C4D3×#4112yes—
443C8—42no—
444C8—64no—
445C2×C3#243111yescharacteristic
446C2×C3#301312yes—
447C2×C3#317812yes—
448C2×C3#322211yescharacteristic
449C2×C3#322311yescharacteristic
450D3—81no—
451D3—161no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—122no—
456C22—122no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22—242no—
461C22—242no—
462C22C2×#88211yescharacteristic
463C22#566811yescharacteristic
464C22#571511yescharacteristic
465C22#576212yes—
466C22#576612yes—
467C4—24no—
468C4—44no—
469C4—81no—
470C4—81no—
471C4—121no—
472C4—122no—
473C4—124no—
474C4—241no—
475C4#555412yes—
476C4#559912yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—241no—
479C2—481no—
480C2#7631311yescharacteristic
481C2#7879512yes—
482C2#8159412yes—
483C2#8176911yescharacteristic
484C2#8180411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1956D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×#5086C2semidirect
3. C2×#4697C2semidirect
5. C3×#1956C2semidirect
6. #70273C2semidirect
8. #70240C2semidirect
9. #70239C2semidirect
10. #69753C2semidirect
11. #69583C2semidirect
51. #1956D3semidirect
62. C3×C4⋊Q8D4semidirect
102. #870D4semidirect
217. C4⋊Q8D12semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#50862
C2×#46972
#743702
C3×#19562
#702732
#702722
#702402
#702392
#697532
#695832
#256123