Finite Group Databasebeta

← #10883530

Subgroups — #10883530

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3037C211yesmaximal, characteristic
2#83314C211yesmaximal, characteristic
3#83276C211yesmaximal, characteristic
4C2×#5030C211yesmaximal, characteristic, complemented
5#72109C212yesmaximal, complemented
6#72103C212yesmaximal, complemented
7#71637C211yesmaximal, characteristic, complemented
8#70528C211yesmaximal, characteristic, complemented
9#70517C211yesmaximal, characteristic, complemented
10#70304C212yesmaximal, complemented
11#70302C212yesmaximal, complemented
12#28179—31nomaximal, Sylow
13C2×C3×#137C2212yes—
14C3×#532C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#951C2212yes—
17C2×#948C2212yes—
18C22×#216C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#464—22no—
21C3×#462—21no—
22C3×#460—21no—
23C3×#423C2211yescharacteristic
24C3×#417C2212yes—
25#5234C2212yes—
26#5232C2212yes—
27C2×#836C2212yes—
28C2×#835C2211yescharacteristic
29C2×#824C2212yes—
30#5090—21no—
31#5075—21no—
32#5054—22no—
33#5030—22no—
34#5030—22no—
35#5011—22no—
36#5011—22no—
37C2×#810C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4924C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×#803C2211yescharacteristic
42#4558C2212yes—
43#4558C2212yes—
44#4554C2212yes—
45#4530C2212yes—
46#3037—31no—
47#3037D311yescharacteristic
48C2×#441—31no—
49#2932—31no—
50#2079—32no—
51#2020—32no—
52#1908—32no—
53#1908—32no—
54#1838—31no—
55#1801—31no—
56#1801—31no—
57C2×C3×#41C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C3×#137—24no—
61C3×#137—42no—
62C2×C3×Q16—22no—
63C2×C3×SD16—22no—
64C2×C3×M16C2312yes—
65C3×C4⋊Q8D411yescharacteristic
66C3×#127D411yescharacteristic
67C3×#124D412yes—
68C3×#121—21no—
69C3×#121—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80#951—24no—
81#951—42no—
82#948—42no—
83#948—42no—
84C2×#216—22no—
85C2×#216—22no—
86C2×#216—22no—
87C2×#216—22no—
88C2×#214—22no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C2×#208—22no—
93C2×#208C2312yes—
94C2×#208C2312yes—
95C22×C3⋊C8—22no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C4×Dic6—21no—
99C4×Dic6—21no—
100C4×Dic6—21no—
101C4×Dic6—21no—
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×#103—22no—
105C3×#102—22no—
106#836—42no—
107#835—22no—
108#835—22no—
109#832—22no—
110#830—24no—
111#824—42no—
112#810—22no—
113#808—22no—
114C4⋊(C3⋊C8)—21no—
115C4⋊(C3⋊C8)—21no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×#137—32no—
121C2×#137C2×D312yes—
122C22×Q16—31no—
123C2×#131—32no—
124#532C2×D311yescharacteristic
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127#464—22no—
128#464—62no—
129#462—21no—
130#462—61no—
131#460—21no—
132#460—61no—
133#441—62no—
134#441—62no—
135#435—62no—
136#435—62no—
137C2×#106—31no—
138#423—31no—
139#423—31no—
140#423C2×D311yescharacteristic
141#417—32no—
142#417C2×D312yes—
143C2×#103—31no—
144C2×#103—31no—
145C2×#101—32no—
146C2×#101—32no—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343—32no—
151#341—32no—
152#341—32no—
153C3×#41—22no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8C2×D412yes—
163C2×C3×Q8#4112yes—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4#4111yescharacteristic
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6#4112yes—
182C3×Q16—44no—
183C3×SD16—44no—
184C3×M16—22no—
185C3×M16—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—41no—
196C3×C42—41no—
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201#216—28no—
202#216—44no—
203#214—44no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206Dic3⋊C4—41no—
207Dic3⋊C4—41no—
208C4×Dic3—41no—
209C4×Dic3—41no—
210#208—22no—
211#208—22no—
212#208—22no—
213#208—22no—
214#208—44no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—44no—
222C2×C3⋊C8C2×D412yes—
223C2×C3⋊C8C2×D412yes—
224C2×#41C22×D312yes—
225C22×Q8—31no—
226C22×Q8C22×D311yescharacteristic
227#137—24no—
228#137—42no—
229#137—64no—
230#137—122no—
231C2×Q16—22no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×Q16—62no—
235C2×Q16—62no—
236C2×SD16—22no—
237C2×SD16—62no—
238#131—122no—
239#131—122no—
240C2×M16—32no—
241C2×M16—32no—
242C2×M16—62no—
243C2×M16C22×D312yes—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8—62no—
247C4⋊Q8#6711yescharacteristic
248#127#6711yescharacteristic
249#124#6712yes—
250#121—21no—
251#121—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—61no—
257C4×Q8—61no—
258C4×Q8—61no—
259C4×Q8—61no—
260C4×D4—21no—
261C4×D4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C42—21no—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#106—62no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—61no—
273C4⋊C8—61no—
274C4⋊C8—61no—
275C4⋊C8—61no—
276#103—22no—
277#103—62no—
278#103—62no—
279#103—62no—
280#102—22no—
281#102—62no—
282#101—122no—
283#101—122no—
284C22⋊C8—64no—
285#97—32no—
286#97—32no—
287C23×C3—22no—
288C23×C3C2×#4111yescharacteristic
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—22no—
292C3×Q8—22no—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—81no—
300C3×D4—81no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—81no—
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—81no—
326Dic6—22no—
327Dic6—22no—
328Dic6—41no—
329Dic6—41no—
330C3×C8—42no—
331C3⋊C8—22no—
332C3⋊C8—22no—
333C3⋊C8—42no—
334C3⋊C8—42no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—32no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8#23712yes—
351C2×Q8C2×#6712yes—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C2×#6712yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4#23711yescharacteristic
362C22×C4#23711yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—44no—
365Q16—68no—
366Q16—124no—
367SD16—44no—
368SD16—124no—
369M16—22no—
370M16—22no—
371M16—62no—
372M16—62no—
373M16—62no—
374M16—62no—
375M16—124no—
376C2×C8—22no—
377C2×C8—32no—
378C2×C8—32no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—124no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—121no—
395C4⋊C4—121no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C42—41no—
399C42—41no—
400C42—121no—
401C42—121no—
402C42D3×D411yescharacteristic
403C42D3×D411yescharacteristic
404C42C2×#6711yescharacteristic
405C42C2×#6711yescharacteristic
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3C2×#12111yescharacteristic
412C22×C3#55311yescharacteristic
413C22×C3#55311yescharacteristic
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4#52511yescharacteristic
422C3×C4#53511yescharacteristic
423C3×C4#54611yescharacteristic
424C3×C4#54611yescharacteristic
425Dic3—42no—
426Dic3—81no—
427C23—22no—
428C23C2×#23711yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—62no—
436Q8—62no—
437Q8—121no—
438Q8—121no—
439D4—22no—
440D4—22no—
441D4—42no—
442D4—42no—
443D4—81no—
444D4—81no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—62no—
455C2×C4—81no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—241no—
460C2×C4#93911yescharacteristic
461C2×C4#93911yescharacteristic
462C2×C4#93911yescharacteristic
463C2×C4#93911yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×#23711yescharacteristic
466C2×C4#100411yescharacteristic
467C2×C4D3×#4111yescharacteristic
468C2×C4#100911yescharacteristic
469C2×C4C22×#6711yescharacteristic
470C8—42no—
471C8—62no—
472C8—62no—
473C8—122no—
474C8—122no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—82no—
478C2×C3#253511yescharacteristic
479C2×C3#306911yescharacteristic
480C2×C3#306911yescharacteristic
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22#561411yescharacteristic
487C22#561811yescharacteristic
488C22C2×#93911yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—81no—
496C4—122no—
497C4—241no—
498C4#560611yescharacteristic
499C4#560711yescharacteristic
500C4#580611yescharacteristic
501C4#582911yescharacteristic
502C3#2817911yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—82no—
506C2#8266611yescharacteristic
507C2#8397811yescharacteristic
508C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3037C2
socleC22×C3C2×#121
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#28179
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
4. C2×#5030C2semidirect
5. #72109C2semidirect
6. #72103C2semidirect
7. #71637C2semidirect
8. #70528C2semidirect
9. #70517C2semidirect
10. #70304C2semidirect
11. #70302C2semidirect
502. C3#28179semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#833142
#832762
C2×#50302
#721092
#721032
#716372
#705282
#705172
#703042
#703022
#281793