Finite Group Databasebeta

← #10877167

Subgroups — #10877167

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83794C211yesmaximal, characteristic
2C2×D3×#107C211yesmaximal, characteristic, complemented
3#78892C212yesmaximal
4C3×#1816C211yesmaximal, characteristic, complemented
5#72016C211yesmaximal, characteristic, complemented
6#70711C211yesmaximal, characteristic, complemented
7#70701C211yesmaximal, characteristic, complemented
8#70625C211yesmaximal, characteristic, complemented
9#70301C212yesmaximal, complemented
10#68990C212yesmaximal, complemented
11#68980C212yesmaximal, complemented
12#27675—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5828C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×D3×C4⋊C4C2212yes—
17C2×C3×#108C2211yescharacteristic
18C2×C3×#107C2211yescharacteristic
19C3×#417C2212yes—
20#5234C2211yescharacteristic
21#5232C2212yes—
22#5127—21no—
23#5127—21no—
24#5126C2212yes—
25#5126C2211yescharacteristic
26#5121C2212yes—
27#5119C2211yescharacteristic
28C8×#67—21no—
29C8×#67—21no—
30C2×#819C2211yescharacteristic
31C2×#818C2211yescharacteristic
32#5101C2211yescharacteristic
33#4964C2212yes—
34C2×#807C2212yes—
35D3×#107—21no—
36D3×#107—21no—
37D3×#107—21no—
38D3×#107—21no—
39#4811—22no—
40#4805—22no—
41#4777—22no—
42#4774—22no—
43C3×#343C2212yes—
44#4558C2212yes—
45#4554C2212yes—
46#4554C2212yes—
47#4530C2212yes—
48#3092—32no—
49#2929—31no—
50C22×#107—31no—
51#2089—32no—
52#2078—32no—
53#1952—31no—
54#1952—31no—
55#1821—32no—
56#1816—31no—
57#1816D311yescharacteristic, complemented
58#1810—31no—
59C2×C3×#41C2311yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3C2311yescharacteristic
62C3×#131—22no—
63C3×#131—22no—
64C2×C3×M16—21no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2311yescharacteristic
67C22×C3×C8—21no—
68C22×C3×C8C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70#946D411yescharacteristic
71#946D411yescharacteristic
72#938D411yescharacteristic
73#938D411yescharacteristic
74#937—21no—
75#935—21no—
76C4×#67—21no—
77C4×#67—21no—
78#926C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—22no—
81C2×#208C2312yes—
82C2×#208C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C2×D3×C8—22no—
85C2×D3×C8—22no—
86C2×D3×C8D411yescharacteristic
87C2×D3×C8D411yescharacteristic
88C2×D3×C8D411yescharacteristic
89C2×D3×C8D411yescharacteristic
90D3×C4⋊C4—22no—
91D3×C4⋊C4—22no—
92D3×C4⋊C4—22no—
93D3×C4⋊C4—22no—
94C3×#108—22no—
95C3×#107—21no—
96C3×#107—21no—
97C3×#103—22no—
98C3×#102—22no—
99#835—22no—
100#832—22no—
101#830—21no—
102#830—21no—
103(C2×D3)⋊C8—21no—
104(C2×D3)⋊C8—21no—
105(C2×D3)⋊C8—22no—
106(C2×D3)⋊C8—22no—
107#819—22no—
108#818—21no—
109#818—21no—
110#818—22no—
111#817—22no—
112#815—21no—
113Dic3⋊C8—21no—
114Dic3⋊C8—21no—
115C8×Dic3—21no—
116#808—22no—
117#807—22no—
118#807—22no—
119#807—22no—
120C2×#131C2×D311yescharacteristic
121C23×C8—31no—
122#525—31no—
123C22×C4⋊C4—32no—
124#484—62no—
125#483—62no—
126#481—62no—
127#480—62no—
128#435—61no—
129#435—61no—
130C8×D4—61no—
131C8×D4—61no—
132C2×#108—31no—
133C2×#108C2×D311yescharacteristic
134#428—32no—
135#428—32no—
136C2×#107—31no—
137C2×#107—32no—
138C2×#107—61no—
139C2×#107—61no—
140C2×#107—61no—
141C2×#107—61no—
142C2×#107C2×D311yescharacteristic
143#424—31no—
144#417—32no—
145#417C2×D312yes—
146#408—31no—
147#407—32no—
148#407—31no—
149#405—31no—
150#343—32no—
151#343—32no—
152#343—32no—
153#343C2×D312yes—
154#341—32no—
155C23×D3#4111yescharacteristic
156C3×#41—21no—
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C2×C3×Q8C2×D411yescharacteristic
161C2×C3×D4—21no—
162C2×C3×D4C2×D411yescharacteristic
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4C2411yescharacteristic
166C2×#67—21no—
167C2×#67—21no—
168C22×Dic3—22no—
169C22×Dic3#4111yescharacteristic
170C2×C4×D3—21no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C4×D3—84no—
177C2×C4×D3C2×D411yescharacteristic
178C2×C4×D3C2×D411yescharacteristic
179C2×C4×D3C2×D411yescharacteristic
180C2×C4×D3C2×D411yescharacteristic
181C3×M16—22no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—22no—
185C3×M16—42no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—42no—
189C2×C3×C8C2×D411yescharacteristic
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8#4112yes—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195#217—21no—
196#217—21no—
197#217—21no—
198(C2×D3)⋊C4—21no—
199(C2×D3)⋊C4—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3C2×D411yescharacteristic
203C4⋊Dic3C2×D411yescharacteristic
204Dic3⋊C4—21no—
205Dic3⋊C4—21no—
206Dic3⋊C4—42no—
207Dic3⋊C4—42no—
208C4×Dic3—21no—
209#208—22no—
210#208—21no—
211#208—22no—
212#208—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C2×D411yescharacteristic
218C2×C3⋊C8C2×D411yescharacteristic
219D3×C8—22no—
220D3×C8—22no—
221D3×C8—22no—
222D3×C8—22no—
223C2×#41C22×D311yescharacteristic
224C23×C4—31no—
225C23×C4—62no—
226#131—22no—
227#131—22no—
228C2×M16—21no—
229C2×M16—32no—
230C2×M16—31no—
231C2×M16C22×D311yescharacteristic
232C2×M16C22×D312yes—
233C22×C8—21no—
234C22×C8—31no—
235C22×C8—31no—
236C22×C8—31no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8—62no—
240C22×C8—62no—
241C22×C8C22×D311yescharacteristic
242#123—61no—
243C22⋊Q8—31no—
244C22⋊Q8—31no—
245#121—31no—
246#121—31no—
247#120—61no—
248C4×D4—61no—
249C4×D4—61no—
250#117—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D312yes—
258#108—22no—
259#108—62no—
260#107—21no—
261#107—21no—
262#107—62no—
263#107—62no—
264#107—62no—
265#107—62no—
266#107—61no—
267#107—61no—
268#106—62no—
269#106—62no—
270C4⋊C8—61no—
271C4⋊C8—61no—
272#103—22no—
273#103—62no—
274#102—22no—
275#102—62no—
276C22⋊C8—61no—
277C22⋊C8—61no—
278C22⋊C8—62no—
279C22⋊C8—62no—
280C22⋊C8—61no—
281C22⋊C8—61no—
282#97—61no—
283C4×C8—61no—
284C23×C3—21no—
285C23×C3C2×#4111yescharacteristic
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3#12111yescharacteristic
291C22×D3#12111yescharacteristic
292C3×Q8—21no—
293C3×Q8—21no—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—41no—
297C3×D4—41no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—82no—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C42−1+411yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309#67—42no—
310#67—42no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—42no—
316C2×Dic3—82no—
317C2×Dic3#12111yescharacteristic
318C2×Dic3#12111yescharacteristic
319C4×D3—21no—
320C4×D3—21no—
321C4×D3—21no—
322C4×D3—21no—
323C4×D3—21no—
324C4×D3—21no—
325C4×D3—21no—
326C4×D3—21no—
327C4×D3—84no—
328C4×D3—84no—
329C3×C8—41no—
330C3×C8—42no—
331C3×C8#12312yes—
332C3×C8#12312yes—
333C3⋊C8—21no—
334C3⋊C8—21no—
335C3⋊C8—41no—
336C3⋊C8—42no—
337C24—31no—
338#41—21no—
339#41—21no—
340#41—21no—
341#41—21no—
342C2×Q8D3×D411yescharacteristic
343C2×D4—21no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4D3×D411yescharacteristic
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—122no—
360C22×C4—122no—
361C22×C4—244no—
362C22×C4C23×D311yescharacteristic
363M16—21no—
364M16—22no—
365M16—21no—
366M16—22no—
367M16—42no—
368M16—62no—
369M16—62no—
370M16—61no—
371M16—61no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—31no—
375C2×C8—31no—
376C2×C8—42no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—61no—
384C2×C8—62no—
385C2×C8#23512yes—
386C2×C8D3×D411yescharacteristic
387C2×C8D3×D411yescharacteristic
388C2×C8#23712yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—31no—
392C4⋊C4—31no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—122no—
398C4⋊C4—122no—
399C22⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C42—61no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3C2×#12111yescharacteristic
411C22×C3#55211yescharacteristic
412C22×C3#55211yescharacteristic
413C2×D3—24no—
414C2×D3—24no—
415C2×D3—21no—
416C2×D3—21no—
417C2×D3—21no—
418C2×D3—21no—
419C2×D3#47212yes—
420C2×D3#47312yes—
421C3×C4—41no—
422C3×C4—82no—
423C3×C4C2×#12311yescharacteristic
424C3×C4#53611yescharacteristic
425C3×C4#54711yescharacteristic
426C3×C4#54711yescharacteristic
427Dic3—21no—
428Dic3—21no—
429Dic3—41no—
430Dic3—41no—
431Dic3—82no—
432C23—21no—
433C23—31no—
434C23—31no—
435C23—62no—
436C23—62no—
437C23—61no—
438C23—61no—
439C23D3×#4111yescharacteristic
440Q8—21no—
441Q8—21no—
442D4—21no—
443D4—21no—
444D4—41no—
445D4—41no—
446D4—122no—
447D4—122no—
448C2×C4—21no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—82no—
467C2×C4—122no—
468C2×C4—244no—
469C2×C4—244no—
470C2×C4—242no—
471C2×C4C2×#23511yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4C2×#23711yescharacteristic
475C2×C4#100711yescharacteristic
476C2×C4#101011yescharacteristic
477C8—41no—
478C8—42no—
479C8—61no—
480C8—61no—
481C8—122no—
482C8—121no—
483C8#88312yes—
484C8#88312yes—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—81no—
488C2×C3#248611yescharacteristic
489C2×C3C2×#47211yescharacteristic
490C2×C3C2×#47311yescharacteristic
491D3—22no—
492D3—22no—
493C22—21no—
494C22—21no—
495C22—32no—
496C22—32no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—64no—
501C22—64no—
502C22—61no—
503C22—61no—
504C22—61no—
505C22—61no—
506C22#552911yescharacteristic
507C22D3×#12111yescharacteristic
508C22#563911yescharacteristic
509C4—41no—
510C4—61no—
511C4—61no—
512C4—82no—
513C4—121no—
514C4—121no—
515C4—242no—
516C4C2×#88311yescharacteristic
517C4#550711yescharacteristic
518C4#561111yescharacteristic
519C4#562811yescharacteristic
520C3#2767511yescharacteristic, complemented, Sylow
521C2—21no—
522C2—21no—
523C2—62no—
524C2—62no—
525C2—81no—
526C2#7391111yescharacteristic
527C2D3×#47211yescharacteristic
528C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1816C2
socleC22×C3C2×#121
prime radical#5828C22
hypercenter#1816D3
hypocenterC3#27675
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. C2×D3×#107C2semidirect
4. C3×#1816C2semidirect
5. #72016C2semidirect
6. #70711C2semidirect
7. #70701C2semidirect
8. #70625C2semidirect
9. #70301C2semidirect
10. #68990C2semidirect
11. #68980C2semidirect
57. #1816D3semidirect
520. C3#27675semidirect

Maximal subgroups

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

subgroupindex
#837942
C2×D3×#1072
#788922
C3×#18162
#720162
#707112
#707012
#706252
#703012
#689902
#689802
#276753