Finite Group Databasebeta

← #10845429

Subgroups — #10845429

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#436C211yesmaximal, characteristic
2#82384C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic
4C2×#4850C211yesmaximal, characteristic, complemented
5#71989C212yesmaximal, complemented
6#71988C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69938C212yesmaximal, complemented
9#69936C212yesmaximal, complemented
10#69777C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#25349—31nomaximal, Sylow
13C22×C3×M16C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#924C2212yes—
16C2×#869C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C3×#436—24no—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C22⋊C8C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22#5221—21no—
23#5221—21no—
24#5212C2212yes—
25#5212C2212yes—
26#5212C2212yes—
27C2×#818C2212yes—
28C2×#817C2212yes—
29C2×#816C2211yescharacteristic
30C2×#816C2212yes—
31C2×#816C2211yescharacteristic
32#4977—21no—
33#4972—21no—
34#4948—22no—
35#4947—22no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4850—24no—
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4551C2212yes—
42#4551C2211yescharacteristic
43#4551C2212yes—
44C2×#477—31no—
45C2×#436D311yescharacteristic
46#2563—31no—
47#2538—31no—
48#2075—32no—
49#2035—32no—
50#1990—31no—
51#1948—31no—
52#1894—32no—
53#1809—32no—
54C4×#103—31no—
55C22×C3×D4D411yescharacteristic
56C23×C3×C4C2312yes—
57C22×Dic6—21no—
58C22×Dic6C2311yescharacteristic
59C2×C3×M16—24no—
60C2×C3×M16—24no—
61C2×C3×M16—42no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4D412yes—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82#924—24no—
83#924—42no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3—21no—
87C2×C4×Dic3—21no—
88#869—22no—
89#869—22no—
90C4×Dic6—22no—
91C4×Dic6—22no—
92C3×C4⋊C8D414yes—
93C3×C22⋊C8—24no—
94C3×C4×C8—22no—
95C3⋊#95—21no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#818—24no—
100#817—42no—
101#816—22no—
102#816—24no—
103#816—22no—
104C22×M16C2×D312yes—
105C2×C4⋊Q8—31no—
106C2×C22⋊Q8—32no—
107C2×C4×Q8—31no—
108C2×C4×D4C2×D311yescharacteristic
109#477—64no—
110#436—24no—
111C2×#107—32no—
112C2×#106—32no—
113C2×C4⋊C8C2×D311yescharacteristic
114C2×#103—32no—
115C2×#103—31no—
116C2×#103—31no—
117C2×C22⋊C8C2×D312yes—
118C2×C4×C8C2×D311yescharacteristic
119#396—62no—
120#393—62no—
121#390—61no—
122#390—61no—
123#389—61no—
124#384—31no—
125#382—61no—
126#380—32no—
127#380—32no—
128#380—32no—
129C4×C4⋊C4—31no—
130#340—31no—
131#340—32no—
132#340—31no—
133#340—32no—
134#340—32no—
135C24×C3C2×D412yes—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C2×Dic6—21no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—42no—
161C2×Dic6—81no—
162C2×Dic6—81no—
163C2×Dic6#4112yes—
164C3×M16—48no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8C2×D414yes—
173C2×C3×C8#4112yes—
174C3×C4⋊C4—22no—
175C3×C22⋊C4—24no—
176C3×C42C2×D412yes—
177C3×C42#4112yes—
178#217—44no—
179#217—82no—
180#217—82no—
181#217—82no—
182C4⋊Dic3—22no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—21no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—81no—
195Dic3⋊C4—44no—
196Dic3⋊C4—42no—
197C4×Dic3—42no—
198C4×Dic3—42no—
199C22×Q8—31no—
200C22×Q8—61no—
201C22×D4D1211yescharacteristic
202C23×C4C22×D312yes—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—42no—
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D312yes—
209C4⋊Q8—62no—
210C4⋊Q8—62no—
211C22⋊Q8—64no—
212C22⋊Q8—122no—
213C4×Q8—62no—
214C4×Q8—62no—
215C4×D4—22no—
216C4×D4—22no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4D1211yescharacteristic
228C2×C22⋊C4—62no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4D1212yes—
233C2×C42—61no—
234C2×C42—61no—
235C2×C42C22×D311yescharacteristic
236#107—64no—
237#106—122no—
238C4⋊C8#6714yes—
239#103—62no—
240#103—62no—
241#103—64no—
242C22⋊C8—24no—
243C4×C8—22no—
244#95—61no—
245#95—62no—
246#95—62no—
247#95—62no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3C22×D411yescharacteristic
256C3×D4—44no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4#12112yes—
267C2×C3×C4#12112yes—
268C2×C3×C4#13712yes—
269C2×C3×C4#13712yes—
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4H(5,𝔽2)11yescharacteristic
275C2×C3×C42−1+411yescharacteristic
276C2×Dic3—22no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294Dic6—22no—
295Dic6—22no—
296Dic6—41no—
297Dic6—41no—
298Dic6—84no—
299C3×C8—24no—
300C3×C8—22no—
301C3×C8—41no—
302C24C2×D1212yes—
303C2×Q8—32no—
304C2×Q8—61no—
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—122no—
310C2×Q8—241no—
311C2×Q8—241no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4C2×D1212yes—
328C22×C4C2×D1211yescharacteristic
329C22×C4C2×D1211yescharacteristic
330C22×C4C23×D311yescharacteristic
331M16—48no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8#23712yes—
340C2×C8C2×#6714yes—
341C4⋊C4—22no—
342C4⋊C4—61no—
343C4⋊C4—62no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—62no—
347C4⋊C4—61no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—62no—
351C4⋊C4—61no—
352C4⋊C4—124no—
353C4⋊C4—122no—
354C4⋊C4—122no—
355C4⋊C4—122no—
356C4⋊C4—241no—
357C22⋊C4—24no—
358C22⋊C4—124no—
359C22⋊C4—242no—
360C22⋊C4—242no—
361C22⋊C4—242no—
362C42—122no—
363C42—122no—
364C42#23512yes—
365C42C2×#6712yes—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3C2×#12111yescharacteristic
374C22×C3#54611yescharacteristic
375C22×C3#54711yescharacteristic
376C22×C3C2×#13711yescharacteristic
377C22×C3C2×#13711yescharacteristic
378C22×C3#57611yescharacteristic
379C22×C3#57711yescharacteristic
380C3×C4—22no—
381C3×C4—42no—
382C3×C4#47014yes—
383C3×C4#52911yescharacteristic
384C3×C4#54211yescharacteristic
385C3×C4#54611yescharacteristic
386C3×C4#54811yescharacteristic
387Dic3—42no—
388Dic3—81no—
389Dic3—82no—
390Dic3—81no—
391Dic3—82no—
392Dic3—81no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23C22×D1211yescharacteristic
401Q8—62no—
402Q8—62no—
403Q8—121no—
404Q8—121no—
405Q8—244no—
406D4—44no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—62no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—122no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4#90912yes—
435C2×C4#90912yes—
436C2×C4#93012yes—
437C2×C4#93012yes—
438C2×C4C2×#23511yescharacteristic
439C2×C4C2×#23711yescharacteristic
440C2×C4#100411yescharacteristic
441C2×C4D3×#4111yescharacteristic
442C2×C4#101011yescharacteristic
443C2×C4C22×#6711yescharacteristic
444C8—22no—
445C8—24no—
446C8—41no—
447C2×C3—82no—
448C2×C3#242711yescharacteristic
449C2×C3C2×#47011yescharacteristic
450C2×C3#306211yescharacteristic
451C2×C3#332911yescharacteristic
452C2×C3#333311yescharacteristic
453C2×C3#335611yescharacteristic
454C2×C3#335711yescharacteristic
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22#555411yescharacteristic
463C22#555511yescharacteristic
464C22C2×#90911yescharacteristic
465C22C2×#90911yescharacteristic
466C22#575111yescharacteristic
467C22#575211yescharacteristic
468C22C2×#93011yescharacteristic
469C4—22no—
470C4—42no—
471C4—122no—
472C4—242no—
473C4—241no—
474C4—241no—
475C4—241no—
476C4—242no—
477C4#513714yes—
478C4#555711yescharacteristic
479C4#556111yescharacteristic
480C4#580611yescharacteristic
481C4#583111yescharacteristic
482C3#2534911yescharacteristic, complemented, Sylow
483C2—82no—
484C2#7953411yescharacteristic
485C2#7953511yescharacteristic
486C2#7956411yescharacteristic
487C2#7956611yescharacteristic
488C2#8240811yescharacteristic
489C2C2×#513711yescharacteristic
490C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#436C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#436D3
hypocenterC3#25349
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
4. C2×#4850C2semidirect
5. #71989C2semidirect
6. #71988C2semidirect
7. #70454C2semidirect
8. #69938C2semidirect
9. #69936C2semidirect
10. #69777C2semidirect
11. C4×#816C2semidirect
482. C3#25349semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4362
#823842
#823762
C2×#48502
#719892
#719882
#704542
#699382
#699362
#697772
C4×#8162
#253493