Finite Group Databasebeta

← #10818075

Subgroups — #10818075

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2950C211yesmaximal, characteristic, complemented
2C2×#5192C211yesmaximal, characteristic
3#82547C212yesmaximal
4#79179C211yesmaximal, characteristic
5#72066C211yesmaximal, characteristic, complemented
6#71658C212yesmaximal, complemented
7#70659C211yesmaximal, characteristic, complemented
8#70348C212yesmaximal
9#70338C212yesmaximal, complemented
10#69894C211yesmaximal, characteristic, complemented
11#69887C211yesmaximal, characteristic
12#23045—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×#519C2212yes—
15C2×#942C2212yes—
16C2×#901C2211yescharacteristic
17#5523C2212yes—
18C3×#441—22no—
19C3×C4×Q16—21no—
20C3×C4×Q16—21no—
21C3×#427C2211yescharacteristic
22C3×#419C2212yes—
23C3×#405C2211yescharacteristic
24#5229C2212yes—
25C2×#836C2212yes—
26C2×#835C2212yes—
27#5192C418yescomplemented
28C2×#825C2211yescharacteristic
29C2×#819C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#5030—22no—
33C4×#216—22no—
34#5026—22no—
35C4×#215—22no—
36#5002C2211yescharacteristic
37#4997C2212yes—
38#4995C2212yes—
39#4995C2212yes—
40#4880—21no—
41D3×#107—21no—
42#4861—22no—
43#4558C2211yescharacteristic
44#4554C2212yes—
45#4532C2212yes—
46#4532C2211yescharacteristic
47C2×#473—31no—
48#2950D311yescharacteristic, complemented
49#2947—32no—
50#2913—31no—
51#1969—31no—
52#1952—31no—
53#1930—32no—
54#1882—32no—
55#1857—32no—
56#1772—31no—
57#1764—31no—
58C22×C3×Q8C2312yes—
59C2×#235C2311yescharacteristic
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C2×C3×Q16—22no—
63C2×C3×Q16C2×C414yescomplemented
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×#117—42no—
71C3×#117C2311yescharacteristic
72C3×#117C2312yes—
73C2×#216—22no—
74#942—24no—
75#942C2×C418yes—
76C2×#215—22no—
77C2×C4⋊Dic3C2312yes—
78C2×C4×Dic3C2311yescharacteristic
79C2×#208C2311yescharacteristic
80C2×#208C2312yes—
81C22×C3⋊C8C2311yescharacteristic
82#901—22no—
83#901C2×C414yes—
84C2×ΓL(1,𝔽25)—21no—
85C2×D3×C8—21no—
86#892—22no—
87D3×C4⋊C4—22no—
88#67⋊C4—42no—
89C4×D12—22no—
90C4×Dic6—22no—
91C3×#107D411yescharacteristic
92C3×#107D411yescharacteristic
93C3×#106D412yes—
94C3×#103—22no—
95C3×#103—22no—
96C3×#97—21no—
97C3×C4×C8—21no—
98#836C2×C418yescomplemented
99#835—22no—
100#835—22no—
101C3⋊#95—42no—
102#825—22no—
103#819C2×C414yescomplemented
104#818—21no—
105#818—21no—
106C8×Dic3—22no—
107#810D411yescharacteristic
108#810D411yescharacteristic
109#809D411yescharacteristic
110#809D411yescharacteristic
111#808—22no—
112#808—22no—
113#807—22no—
114#807—22no—
115#803—22no—
116C4×C3⋊C8—22no—
117C2×#137—32no—
118C22×Q16C2×D311yescharacteristic
119C2×#131—31no—
120#520—32no—
121#519C2×D312yes—
122#473—38no—
123#441—22no—
124#441—62no—
125#440—62no—
126C4×Q16—21no—
127C4×Q16—21no—
128C4×Q16—62no—
129C4×SD16—62no—
130C2×#108—31no—
131#428—32no—
132#428—32no—
133#428—62no—
134#427—61no—
135#427C2×D311yescharacteristic
136C2×#107—31no—
137C2×#107—61no—
138C2×#104—31no—
139#419C2×D312yes—
140#416—31no—
141C2×#103—32no—
142C2×#101—32no—
143#405—32no—
144#405C2×D311yescharacteristic
145C2×C4×C8—31no—
146#391—32no—
147#343—32no—
148#341—31no—
149#339—31no—
150#339—32no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C22×C414yes—
157C22×C3×C4—22no—
158C22×C3×C4C2411yescharacteristic
159C2×#67—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162#235—21no—
163#235—21no—
164#235C22×C412yes—
165#235#4112yes—
166C2×D12C2×D411yescharacteristic
167C2×C4×D3—21no—
168C2×C4×D3—42no—
169C2×Dic6C2×D411yescharacteristic
170C3×Q16—24no—
171C3×Q16—24no—
172C3×M16—21no—
173C3×M16—21no—
174C2×C3×C8—21no—
175C2×C3×C8C22×C412yes—
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8#4112yes—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4C2×D411yescharacteristic
182C3×C4⋊C4C2×D411yescharacteristic
183C3×C4⋊C4C2×D412yes—
184C3×C4⋊C4C2×D412yes—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—21no—
187C3×C42—22no—
188C3×C42—21no—
189C3×C42—42no—
190C3×C42—42no—
191#216—28no—
192#215—28no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196Dic3⋊C4—42no—
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3—42no—
200#208—21no—
201#208—22no—
202#208C22×C414yes—
203#208C22×C412yes—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8C2×D411yescharacteristic
209C2×C3⋊C8C2×D411yescharacteristic
210ΓL(1,𝔽25)—24no—
211D3×C8—24no—
212C2×#41—31no—
213C22×Q8C22×D312yes—
214#137—38no—
215#137—64no—
216C2×Q16—22no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—62no—
220C2×Q16C4×D314yescomplemented
221C2×SD16—62no—
222#131—34no—
223#131—62no—
224C2×M16—31no—
225C2×M16—32no—
226C2×M16—61no—
227C2×M16C22×D311yescharacteristic
228C22×C8—31no—
229C22×C8—61no—
230C22×C8C22×D311yescharacteristic
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—62no—
236C4×D4—62no—
237C4×D4—122no—
238#117—42no—
239#117—62no—
240#117C22×D311yescharacteristic
241#117C22×D312yes—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C42—31no—
245#108—34no—
246#107—61no—
247#107—61no—
248#107—62no—
249#107—62no—
250#107#6711yescharacteristic
251#107#6711yescharacteristic
252#106—62no—
253#106—62no—
254#106#6712yes—
255#104—62no—
256#103—22no—
257#103—22no—
258#103—31no—
259#103—31no—
260#103—62no—
261#103—62no—
262#102—31no—
263#102—31no—
264#101—38no—
265#97—21no—
266#97—62no—
267C4×C8—21no—
268C4×C8—62no—
269C4×C8—62no—
270#95—122no—
271C23×C3H(5,𝔽2)11yescharacteristic
272C22×D3—21no—
273C3×Q8—24no—
274C3×Q8—24no—
275C3×Q8—42no—
276C3×Q8—42no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4C23×C411yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290#67—22no—
291#67—22no—
292C2×Dic3—21no—
293C2×Dic3—22no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—82no—
298D12—21no—
299D12C4×D412yes—
300C4×D3—22no—
301C4×D3—22no—
302C4×D3—162no—
303Dic6—21no—
304Dic6C4×D412yes—
305C3×C8—41no—
306C3×C8C4×D412yes—
307C3×C8C4×D412yes—
308C3⋊C8—22no—
309C3⋊C8—24no—
310C3⋊C8—21no—
311C3⋊C8—21no—
312#41—32no—
313#41—32no—
314#41—61no—
315#41—61no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—31no—
322C2×Q8C2×C4×D314yes—
323C2×D4—31no—
324C2×D4—61no—
325C22×C4—22no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—122no—
330C22×C4C23×D311yescharacteristic
331Q16—24no—
332Q16—24no—
333Q16—68no—
334SD16—68no—
335M16—21no—
336M16—21no—
337M16—32no—
338M16—34no—
339M16—61no—
340M16—62no—
341M16—64no—
342C2×C8—21no—
343C2×C8—31no—
344C2×C8—31no—
345C2×C8—61no—
346C2×C8—62no—
347C2×C8—61no—
348C2×C8—61no—
349C2×C8—64no—
350C2×C8C2×C4×D312yes—
351C2×C8#23512yes—
352C2×C8C2×#6711yescharacteristic
353C2×C8C2×#6711yescharacteristic
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—122no—
359C4⋊C4D3×D411yescharacteristic
360C4⋊C4D3×D411yescharacteristic
361C4⋊C4C2×#6712yes—
362C4⋊C4C2×#6712yes—
363C22⋊C4—21no—
364C22⋊C4—22no—
365C22⋊C4—122no—
366C42—22no—
367C42—21no—
368C42—42no—
369C42—42no—
370C42—61no—
371C42—61no—
372C42—122no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3#52011yescharacteristic
376C22×C3D4211yescharacteristic
377C22×C3#55311yescharacteristic
378C2×D3—22no—
379C2×D3—41no—
380C2×D3—41no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—41no—
384C3×C4—42no—
385C3×C4—41no—
386C3×C4C2×C4×D411yescharacteristic
387C3×C4C2×C4×D411yescharacteristic
388C3×C4#54711yescharacteristic
389C3×C4#54811yescharacteristic
390Dic3—22no—
391Dic3—42no—
392Dic3—82no—
393C23—61no—
394C23#100411yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—32no—
398Q8—42no—
399Q8—42no—
400Q8—61no—
401D4—32no—
402D4—62no—
403D4—62no—
404D4—61no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—61no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—242no—
420C2×C4—482no—
421C2×C4C22×C4×D311yescharacteristic
422C2×C4C2×#23511yescharacteristic
423C2×C4C2×D3×D411yescharacteristic
424C2×C4D3×#4111yescharacteristic
425C2×C4#101011yescharacteristic
426C2×C4C22×#6711yescharacteristic
427C8—41no—
428C8—62no—
429C8—64no—
430C8—61no—
431C8—61no—
432C8C4×#6712yes—
433C8C4×#6712yes—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3#236111yescharacteristic
437C2×C3#329011yescharacteristic
438C2×C3#329311yescharacteristic
439D3—42no—
440C22—21no—
441C22—21no—
442C22—62no—
443C22—121no—
444C22—121no—
445C22#554611yescharacteristic
446C22#555811yescharacteristic
447C22D4×#6711yescharacteristic
448C4—41no—
449C4—42no—
450C4—42no—
451C4—41no—
452C4—44no—
453C4—62no—
454C4—122no—
455C4—242no—
456C4C4×D3×D411yescharacteristic
457C4#555111yescharacteristic
458C4C2×C4×#6711yescharacteristic
459C4#583111yescharacteristic
460C3#2304511yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—122no—
464C2#8239911yescharacteristic
465C2#8401811yescharacteristic
466C2#8402011yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2950C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#2950D3
hypocenterC3#23045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2950

Complements

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

NH ≅ G/Ntype
1. C3×#2950C2semidirect
5. #72066C2semidirect
6. #71658C2semidirect
7. #70659C2semidirect
9. #70338C2semidirect
10. #69894C2semidirect
27. #5192C4semidirect
48. #2950D3semidirect
63. C2×C3×Q16C2×C4semidirect
98. #836C2×C4semidirect
103. #819C2×C4semidirect
220. C2×Q16C4×D3semidirect
460. C3#23045semidirect

Maximal subgroups

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

subgroupindex
C3×#29502
C2×#51922
#825472
#791792
#720662
#716582
#706592
#703482
#703382
#698942
#698872
#230453