Finite Group Databasebeta

← #10698550

Subgroups — #10698550

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83188C211yesmaximal, characteristic
2C3×#1957C211yesmaximal, characteristic, complemented
3C2×#4544C211yesmaximal, characteristic, complemented
4C2×#4459C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic, complemented
6#69568C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10409—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#940—21no—
11C2×#938—21no—
12C2×#938—21no—
13C2×C4×D12C2211yescharacteristic
14C2×C3×#102—22no—
15C2×C3×C4×C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17#5223—22no—
18#5221—21no—
19C2×#809—21no—
20C2×#808—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4544—22no—
25#4529—21no—
26#4529—21no—
27#4459—22no—
28#2541—31no—
29#1957D311yescharacteristic, complemented
30#1948—31no—
31#1802—31no—
32C2×#331—31no—
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4D412yes—
38C22×#67—22no—
39C22×#67—22no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×C3×C8—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52C2×#217—22no—
53#940—42no—
54#940—42no—
55#938—42no—
56#938—42no—
57#938—42no—
58#938—42no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4×Dic3—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C3×#102—44no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3⋊#95—21no—
74#809—42no—
75#808—42no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yescomplemented
80C2×#127—61no—
81C2×#127C2×D311yescharacteristic
82C2×#121—61no—
83C2×#121—61no—
84C2×C4×D4—31no—
85C2×#106—61no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8—31no—
88C2×#102—22no—
89C2×#102—61no—
90C2×C4×C8C2×D311yescharacteristic
91#392—62no—
92#390—61no—
93#384C2×D311yescharacteristic
94C4×C4⋊C4—31no—
95#340—61no—
96#340—61no—
97#331—62no—
98D4⋊C8—62no—
99C24×C3—22no—
100C23×D3—21no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—44no—
122C2×#67—44no—
123C2×#67—44no—
124C2×#67—44no—
125C2×#67—82no—
126C2×#67—82no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12SD1612yes—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152C3×C42#4111yescharacteristic
153C3×C42#4111yescharacteristic
154#217—44no—
155#217—44no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3D812yes—
163C4⋊Dic3SD1612yes—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—31no—
175C22×D4—62no—
176C22×D4—62no—
177C22×D4#6712yes—
178C23×C4—61no—
179C22×C8—21no—
180C22×C8—61no—
181C22×C8—61no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127—21no—
186#127—122no—
187#127—122no—
188#121—122no—
189#121—122no—
190#121—122no—
191#121—122no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—61no—
205C2×C42—61no—
206C2×C42C22×D311yescharacteristic
207#106—122no—
208C4⋊C8—32no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212#102—44no—
213#102—122no—
214C4×C8—21no—
215C4×C8—21no—
216#95—61no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3#12011yescharacteristic
225C22×D3—22no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C3×D4—44no—
233C3×D4—82no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×C3×C4#12311yescharacteristic
252C2×C3×C4#12411yescharacteristic
253C2×C3×C4C2×D811yescharacteristic
254C2×C3×C4C2×SD1611yescharacteristic
255C2×C3×C4C2×SD1611yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261#67—88no—
262#67—88no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272D12—22no—
273D12—22no—
274D12—41no—
275D12—41no—
276C4×D3—81no—
277C4×D3—81no—
278C4×D3—81no—
279C4×D3—81no—
280C3×C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—22no—
284C24—61no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—32no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—124no—
308C2×D4—124no—
309C2×D4—124no—
310C2×D4—124no—
311C2×D4—242no—
312C2×D4—242no—
313C22×C4—41no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—62no—
330C2×C8—121no—
331C2×C8—121no—
332C2×C8—121no—
333C2×C8—121no—
334C2×C8—121no—
335C4⋊C4—32no—
336C4⋊C4—32no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—61no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C22⋊C4—124no—
347C22⋊C4—124no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C42—121no—
351C42—121no—
352C42#23511yescharacteristic
353C42#23511yescharacteristic
354C42D3×D411yescharacteristic
355C42#23711yescharacteristic
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3#39411yescharacteristic
364C22×C3#44711yescharacteristic
365C22×C3#44911yescharacteristic
366C22×C3#45011yescharacteristic
367C22×C3#45011yescharacteristic
368C22×C3#45311yescharacteristic
369C22×C3#45511yescharacteristic
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—81no—
374C2×D3—81no—
375C2×D3—81no—
376C2×D3—81no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—161no—
380C3×C4#45911yescharacteristic
381C3×C4#46011yescharacteristic
382C3×C4#46511yescharacteristic
383C3×C4#46911yescharacteristic
384C3×C4#48111yescharacteristic
385C3×C4#48211yescharacteristic
386C3×C4#48611yescharacteristic
387C3×C4#49011yescharacteristic
388Dic3—42no—
389Dic3—42no—
390Dic3—81no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—62no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—121no—
404C23—121no—
405C23#93711yescharacteristic
406D4—44no—
407D4—62no—
408D4—62no—
409D4—82no—
410D4—81no—
411D4—81no—
412D4—81no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—121no—
419D4—121no—
420D4—248no—
421D4—248no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4—241no—
439C2×C4—241no—
440C2×C4#88311yescharacteristic
441C2×C4#88411yescharacteristic
442C2×C4#88411yescharacteristic
443C2×C4#88511yescharacteristic
444C2×C4D3×D811yescharacteristic
445C2×C4#91111yescharacteristic
446C2×C4#91111yescharacteristic
447C2×C4#91211yescharacteristic
448C2×C4D3×SD1611yescharacteristic
449C2×C4D3×SD1611yescharacteristic
450C2×C4#91411yescharacteristic
451C2×C4#91411yescharacteristic
452C2×C4#92911yescharacteristic
453C2×C4#93911yescharacteristic
454C8—42no—
455C8—122no—
456C8—241no—
457C2×C3—162no—
458C2×C3#162111yescharacteristic
459C2×C3#162911yescharacteristic
460C2×C3#163911yescharacteristic
461C2×C3#164211yescharacteristic
462C2×C3#203411yescharacteristic
463C2×C3#203711yescharacteristic
464C2×C3#204111yescharacteristic
465D3—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—241no—
477C22—241no—
478C22—241no—
479C22—241no—
480C22#495911yescharacteristic
481C22#506111yescharacteristic
482C22#507711yescharacteristic
483C22#515611yescharacteristic
484C22#515811yescharacteristic
485C22#516911yescharacteristic
486C22#516911yescharacteristic
487C4—21no—
488C4—21no—
489C4—122no—
490C4—122no—
491C4—161no—
492C4—241no—
493C4#476011yescharacteristic
494C4#476211yescharacteristic
495C4#476411yescharacteristic
496C4#477711yescharacteristic
497C4#477811yescharacteristic
498C4#478211yescharacteristic
499C4#478411yescharacteristic
500C4#478611yescharacteristic
501C3#1040911yescharacteristic, complemented, Sylow
502C2—162no—
503C2—242no—
504C2#6966511yescharacteristic
505C2#7020711yescharacteristic
506C2#7021111yescharacteristic
507C2#7119811yescharacteristic
508C2#7119911yescharacteristic
509C2#7126611yescharacteristic
510C2#7128511yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1957C2
socleC23×C3#120
prime radical#83188C2
hypercenter#1957D3
hypocenterC3#10409
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C3×#1957C2semidirect
3. C2×#4544C2semidirect
4. C2×#4459C2semidirect
5. #69584C2semidirect
6. #69568C2semidirect
7. #67655C2semidirect
29. #1957D3semidirect
79. C4⋊(C3⋊C8)D4semidirect
501. C3#10409semidirect

Maximal subgroups

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

subgroupindex
#831882
C3×#19572
C2×#45442
C2×#44592
#695842
#695682
#676552
#104093