Finite Group Databasebeta

← #10877170

Subgroups — #10877170

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82139C211yesmaximal, characteristic
2C22×#807C211yesmaximal, characteristic, complemented
3#78626C211yesmaximal, characteristic
4#78622C211yesmaximal, characteristic
5C3×#1821C211yesmaximal, characteristic, complemented
6#69982C211yesmaximal, characteristic, complemented
7#69973C211yesmaximal, characteristic, complemented
8#69010C212yesmaximal, complemented
9#69001C212yesmaximal, complemented
10#68985C212yesmaximal, complemented
11#68980C212yesmaximal, complemented
12#27675—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14#5791C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C23×C3⋊C8C2211yescharacteristic
17C2×#907C2211yescharacteristic
18C3×#407C2211yescharacteristic
19C3×#407C2211yescharacteristic
20#5207C2211yescharacteristic
21#5133C2212yes—
22#5129C2211yescharacteristic
23C2×#822C2212yes—
24#5122C2211yescharacteristic
25#5121C2212yes—
26#5119C2211yescharacteristic
27#4971—21no—
28#4970—21no—
29#4968—21no—
30#4967—21no—
31#4966C2212yes—
32#4964C2212yes—
33C2×#807—21no—
34C2×#807—21no—
35C2×#807—21no—
36C2×#807—21no—
37C2×#807C2212yes—
38C2×#807C2212yes—
39#4736—21no—
40#4735—21no—
41#4730—21no—
42#4730—21no—
43#67⋊C8—21no—
44#67⋊C8—21no—
45#4722—21no—
46#4721—21no—
47C3×#343C2212yes—
48C3×#343C2212yes—
49#4558C2212yes—
50#4554C2212yes—
51#4554C2212yes—
52#3092—31no—
53#3092—31no—
54#2929—31no—
55C22×#107—31no—
56#2089—32no—
57#2078—32no—
58#1952—32no—
59#1821—31no—
60#1821D311yescharacteristic, complemented
61#1816—32no—
62#1810—31no—
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C23×Dic3—21no—
66C2×#235C2311yescharacteristic
67C2×C3×M16C2312yes—
68C2×C3×M16C2311yescharacteristic
69C2×C3×M16C2311yescharacteristic
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76#953—21no—
77#930D411yescharacteristic
78#930D411yescharacteristic
79#929—21no—
80C4×#67—21no—
81C4×#67—21no—
82#926C2311yescharacteristic
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2312yes—
89#924D411yescharacteristic
90#924D411yescharacteristic
91C2×#208C2312yes—
92C22×C3⋊C8—24no—
93C22×C3⋊C8D411yescharacteristic
94C22×C3⋊C8D411yescharacteristic
95C22×C3⋊C8D411yescharacteristic
96C22×C3⋊C8D411yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#907—24no—
100C2×ΓL(1,𝔽25)—21no—
101C2×D3×C8—21no—
102C3×C22⋊C8—21no—
103C3×C22⋊C8—21no—
104C3×C22⋊C8—21no—
105C3×C22⋊C8—21no—
106#830—21no—
107#830—21no—
108#822—24no—
109#821—22no—
110(C2×D3)⋊C8—21no—
111(C2×D3)⋊C8—21no—
112#818—22no—
113#817—22no—
114#816—22no—
115#815—21no—
116Dic3⋊C8—21no—
117Dic3⋊C8—21no—
118C8×Dic3—21no—
119#810—22no—
120#809—22no—
121#808—22no—
122#807—22no—
123#807—22no—
124#807—22no—
125#807—22no—
126#807—22no—
127C2×#131—31no—
128C23×C8—31no—
129#525—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4C2×D311yescharacteristic
132#484—61no—
133#484—61no—
134#483—61no—
135#483—61no—
136#481—61no—
137#481—61no—
138#480—61no—
139#480—61no—
140#435—61no—
141#435—61no—
142C8×D4—61no—
143C8×D4—61no—
144C2×#108—32no—
145#428—32no—
146#428—32no—
147C2×#107—32no—
148C2×#107—32no—
149C2×#107—61no—
150C2×#107—61no—
151C2×#107—61no—
152C2×#107—61no—
153#424—31no—
154#417—32no—
155#417—32no—
156#408—31no—
157#407—31no—
158#407C2×D311yescharacteristic
159#407C2×D311yescharacteristic
160#405—31no—
161#343—32no—
162#343—32no—
163#343C2×D312yes—
164#343C2×D312yes—
165#341—32no—
166C24×C3#4111yescharacteristic
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—21no—
180C2×#67—21no—
181C2×#67—21no—
182C22×Dic3—22no—
183C22×Dic3—41no—
184C22×Dic3—41no—
185C22×Dic3—82no—
186#235—22no—
187#235—22no—
188C2×D12C2×D411yescharacteristic
189C2×C4×D3—21no—
190C2×Dic6C2×D411yescharacteristic
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202#217—21no—
203#217—21no—
204#217—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—41no—
210C4⋊Dic3—41no—
211C4⋊Dic3C2×D411yescharacteristic
212C4⋊Dic3C2×D411yescharacteristic
213Dic3⋊C4—21no—
214Dic3⋊C4—21no—
215C4×Dic3—21no—
216#208—22no—
217#208—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C2×D411yescharacteristic
225C2×C3⋊C8C2×D411yescharacteristic
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228C2×C3⋊C8#4114yes—
229ΓL(1,𝔽25)—42no—
230D3×C8—42no—
231C2×#41—31no—
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D311yescharacteristic
235#131—64no—
236C2×M16—32no—
237C2×M16—61no—
238C2×M16C22×D311yescharacteristic
239C2×M16C22×D312yes—
240C2×M16C22×D311yescharacteristic
241C22×C8—31no—
242C22×C8—31no—
243C22×C8—31no—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—61no—
248C22×C8—64no—
249#123—61no—
250C22⋊Q8—31no—
251C22⋊Q8—31no—
252#121—31no—
253#121—31no—
254#120—61no—
255C4×D4—61no—
256C4×D4—61no—
257#117—31no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D312yes—
270#108—64no—
271#107—62no—
272#107—62no—
273#107—62no—
274#107—62no—
275#107—62no—
276#107—62no—
277#106—62no—
278#106—62no—
279C4⋊C8—61no—
280C4⋊C8—61no—
281#103—62no—
282#103—62no—
283#102—62no—
284#102—62no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—61no—
290C22⋊C8—61no—
291C22⋊C8—61no—
292C22⋊C8—61no—
293#97—61no—
294C4×C8—61no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3#12111yescharacteristic
299C23×C3#12111yescharacteristic
300C23×C3C2×#4111yescharacteristic
301C22×D3—21no—
302C2×C3×C4—22no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—42no—
310C2×C3×C4—82no—
311C2×C3×C4—82no—
312C2×C3×C4—82no—
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4112yes—
318C2×C3×C42−1+412yes—
319#67—42no—
320#67—42no—
321#67—42no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—42no—
326C2×Dic3—82no—
327C2×Dic3—82no—
328C2×Dic3—82no—
329D12—22no—
330C4×D3—41no—
331C4×D3—41no—
332Dic6—22no—
333C3×C8—41no—
334C3×C8—41no—
335C3×C8—42no—
336C3⋊C8—21no—
337C3⋊C8—21no—
338C3⋊C8—42no—
339C3⋊C8#12312yes—
340C3⋊C8#12312yes—
341C24#23511yescharacteristic
342#41—62no—
343#41—62no—
344C2×Q8—31no—
345C2×D4—31no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—82no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—242no—
360C22×C4#23511yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—62no—
371M16—62no—
372M16—122no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—31no—
377C2×C8—31no—
378C2×C8—34no—
379C2×C8—31no—
380C2×C8—31no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—122no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—121no—
399C4⋊C4—121no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C42—61no—
406C22×C3—28no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3#47212yes—
414C22×C3#47312yes—
415C22×C3C2×#12111yescharacteristic
416C22×C3#55212yes—
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—41no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—82no—
423C3×C4C2×#12311yescharacteristic
424C3×C4#53611yescharacteristic
425C3×C4#54712yes—
426Dic3—41no—
427Dic3—41no—
428Dic3—41no—
429Dic3—82no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—61no—
434C23#88411yescharacteristic
435C23#88411yescharacteristic
436C23C2×#23511yescharacteristic
437Q8—62no—
438D4—62no—
439D4—122no—
440D4—122no—
441D4—122no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—242no—
460C2×C4—242no—
461C2×C4—242no—
462C2×C4#88411yescharacteristic
463C2×C4#88411yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×D3×D411yescharacteristic
466C2×C4D3×#4112yes—
467C2×C4#101012yes—
468C8—32no—
469C8—32no—
470C8—41no—
471C8—41no—
472C8—42no—
473C8—61no—
474C8—61no—
475C8—122no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3#248611yescharacteristic
480C2×C3C2×#47211yescharacteristic
481C2×C3C2×#47311yescharacteristic
482D3—81no—
483C22—28no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22#474912yes—
494C22#475512yes—
495C22C2×#88411yescharacteristic
496C22#553112yes—
497C4—21no—
498C4—21no—
499C4—82no—
500C4—121no—
501C4—121no—
502C4—121no—
503C4—242no—
504C4#564811yescharacteristic
505C4D3×#12311yescharacteristic
506C4#565512yes—
507C3#2767511yescharacteristic, complemented, Sylow
508C2—22no—
509C2—22no—
510C2—22no—
511C2—241no—
512C2#7407511yescharacteristic
513C2C2×#474911yescharacteristic
514C2C2×#475511yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1821C2
socleC22×C3C2×#121
prime radical#5791C22
hypercenter#1821D3
hypocenterC3#27675
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. C22×#807C2semidirect
5. C3×#1821C2semidirect
6. #69982C2semidirect
7. #69973C2semidirect
8. #69010C2semidirect
9. #69001C2semidirect
10. #68985C2semidirect
11. #68980C2semidirect
60. #1821D3semidirect
507. C3#27675semidirect

Maximal subgroups

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

subgroupindex
#821392
C22×#8072
#786262
#786222
C3×#18212
#699822
#699732
#690102
#690012
#689852
#689802
#276753