Finite Group Databasebeta

← #10818084

Subgroups — #10818084

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
1C2×#5192C211yesmaximal, characteristic
2#83874C211yesmaximal, characteristic, complemented
3#79146C211yesmaximal, characteristic
4#78774C212yesmaximal
5C3×#1969C211yesmaximal, characteristic, complemented
6#70625C211yesmaximal, characteristic, complemented
7#70263C212yesmaximal, complemented
8#69789C211yesmaximal, characteristic
9#69779C211yesmaximal, characteristic, complemented
10#69693C212yesmaximal
11#69672C212yesmaximal, complemented
12#23045—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14#5812C2212yes—
15C2×#942C2212yes—
16C2×#901C2211yescharacteristic
17#5504C2212yes—
18C2×C3×#107C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C3×#391C2212yes—
22C2×#836C2212yes—
23#5225C2212yes—
24#5192C418yescomplemented
25#5184—22no—
26Dic3×Q16—21no—
27Dic3×Q16—21no—
28#5113C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#5106C2211yescharacteristic
31#5101C2212yes—
32#5101C2211yescharacteristic
33#4964C2212yes—
34#4964C2212yes—
35C2×#805C2211yescharacteristic
36#4881—22no—
37#4880—21no—
38D3×#107—21no—
39#4817—22no—
40#4790—22no—
41#4789—22no—
42#4788—22no—
43#4552C2211yescharacteristic
44#4530C2212yes—
45#4523C2211yescharacteristic
46#4523C2212yes—
47C2×#473—31no—
48#2950—31no—
49#2947—32no—
50#2913—31no—
51#1969D311yescharacteristic, complemented
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
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42C2311yescharacteristic
68Q8×Dic3—22no—
69Q8×Dic3—22no—
70Q8×Dic3—22no—
71Q8×Dic3—22no—
72C2×#216—22no—
73#942—24no—
74#942C2×C418yes—
75C2×#215—22no—
76C4×#67—42no—
77#926—42no—
78#926C2311yescharacteristic
79#926C2312yes—
80C2×#208C2311yescharacteristic
81C2×#208C2311yescharacteristic
82C2×#208C2312yes—
83#901—22no—
84#901C2×C414yes—
85C2×ΓL(1,𝔽25)—21no—
86C2×D3×C8—21no—
87#893—22no—
88#892—22no—
89D3×C4⋊C4—22no—
90#887—22no—
91C3×#107—21no—
92C3×#107—21no—
93C3×#103—22no—
94C3×#103—22no—
95C3×C4×C8—22no—
96C3×#95—42no—
97#836C2×C418yescomplemented
98#835—22no—
99#835—22no—
100#821D411yescharacteristic
101#821D411yescharacteristic
102#819C2×C414yescomplemented
103#818D411yescharacteristic
104#818D411yescharacteristic
105#817D412yes—
106#816D411yescharacteristic
107#816D411yescharacteristic
108#815—21no—
109#815—22no—
110C8×Dic3—21no—
111C8×Dic3—22no—
112#808—22no—
113#808—22no—
114#807—22no—
115#807—22no—
116#805—22no—
117C2×#137—32no—
118C22×Q16C2×D311yescharacteristic
119C2×#131—31no—
120#520—32no—
121#519—32no—
122#473—38no—
123#441—62no—
124#441—62no—
125#440—62no—
126C4×Q16—62no—
127C4×Q16—61no—
128C4×Q16—61no—
129C4×SD16—62no—
130C2×#108—31no—
131#428—32no—
132#428—32no—
133#428—62no—
134#427—31no—
135#427—61no—
136C2×#107—61no—
137C2×#107C2×D311yescharacteristic
138C2×#104—31no—
139#419—32no—
140#416—31no—
141C2×#103C2×D312yes—
142C2×#101—32no—
143#405—32no—
144#405—31no—
145C2×C4×C8C2×D311yescharacteristic
146#391C2×D312yes—
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—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2411yescharacteristic
161C2×#67—21no—
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—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8C22×C412yes—
175C2×C3×C8C2×D411yescharacteristic
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8#4112yes—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C42—21no—
183C3×C42—21no—
184#217—21no—
185#217—22no—
186#216—28no—
187#215—28no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—42no—
190C4⋊Dic3—42no—
191C4⋊Dic3C2×D411yescharacteristic
192C4⋊Dic3C2×D412yes—
193C4⋊Dic3C2×D411yescharacteristic
194C4⋊Dic3C2×D412yes—
195Dic3⋊C4—42no—
196C4×Dic3—22no—
197C4×Dic3—21no—
198C4×Dic3—42no—
199C4×Dic3—42no—
200C4×Dic3—42no—
201#208—21no—
202#208—22no—
203#208—21no—
204#208—21no—
205#208C22×C414yes—
206#208C22×C412yes—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210ΓL(1,𝔽25)—24no—
211D3×C8—24no—
212C2×#41—31no—
213C22×Q8C22×D312yes—
214#137—38no—
215#137—64no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—22no—
219C2×Q16—62no—
220C2×Q16C4×D314yescomplemented
221C2×SD16—62no—
222#131—34no—
223#131—62no—
224C2×M16—31no—
225C2×M16—32no—
226C2×M16—31no—
227C2×M16—61no—
228C22×C8—61no—
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×Q8—62no—
235C4×Q8—62no—
236C4×D4—62no—
237C4×D4—122no—
238#117—31no—
239#117—32no—
240#117—62no—
241#117—122no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4C22×D312yes—
244C2×C42C22×D311yescharacteristic
245#108—34no—
246#107—21no—
247#107—21no—
248#107—31no—
249#107—31no—
250#107—62no—
251#107—62no—
252#106—32no—
253#106—62no—
254#106—62no—
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—62no—
266#97—61no—
267C4×C8—22no—
268C4×C8—62no—
269C4×C8—61no—
270#95—42no—
271C23×C3H(5,𝔽2)11yescharacteristic
272C22×D3—21no—
273C3×Q8—24no—
274C3×Q8—24no—
275C3×Q8—42no—
276C3×Q8—42no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—82no—
285C2×C3×C4C23×C411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291#67—22no—
292#67—22no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298D12—21no—
299D12C4×D412yes—
300C4×D3—22no—
301C4×D3—22no—
302C4×D3—162no—
303Dic6—21no—
304Dic6C4×D412yes—
305C3×C8—21no—
306C3×C8—21no—
307C3×C8C4×D412yes—
308C3×C8C4×D412yes—
309C3⋊C8—22no—
310C3⋊C8—24no—
311C3⋊C8—41no—
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—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—61no—
329C22×C4—122no—
330C22×C4C23×D311yescharacteristic
331Q16—24no—
332Q16—24no—
333Q16—68no—
334SD16—68no—
335M16—32no—
336M16—34no—
337M16—61no—
338M16—62no—
339M16—64no—
340M16—61no—
341M16—61no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—62no—
345C2×C8—61no—
346C2×C8—64no—
347C2×C8—61no—
348C2×C8C2×C4×D312yes—
349C2×C8D3×D411yescharacteristic
350C2×C8D3×D411yescharacteristic
351C2×C8D3×D411yescharacteristic
352C2×C8D3×D411yescharacteristic
353C2×C8#23712yes—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—31no—
357C4⋊C4—31no—
358C4⋊C4—32no—
359C4⋊C4—32no—
360C4⋊C4—122no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C22⋊C4—61no—
364C22⋊C4—62no—
365C22⋊C4—122no—
366C42—21no—
367C42—21no—
368C42—62no—
369C42—61no—
370C42—122no—
371C42—122no—
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—42no—
382C3×C4—44no—
383C3×C4—82no—
384C3×C4C2×C4×D411yescharacteristic
385C3×C4C2×C4×D411yescharacteristic
386C3×C4#54711yescharacteristic
387C3×C4#54811yescharacteristic
388Dic3—22no—
389Dic3—42no—
390Dic3—41no—
391Dic3—41no—
392Dic3—42no—
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—22no—
406C2×C4—24no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—61no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—62no—
419C2×C4—82no—
420C2×C4—482no—
421C2×C4C22×C4×D311yescharacteristic
422C2×C4C2×D3×D411yescharacteristic
423C2×C4C2×D3×D411yescharacteristic
424C2×C4C2×#23711yescharacteristic
425C2×C4D3×#4111yescharacteristic
426C2×C4#101011yescharacteristic
427C8—21no—
428C8—21no—
429C8—62no—
430C8—64no—
431C8—121no—
432C8#67⋊C412yes—
433C8#67⋊C412yes—
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#561411yescharacteristic
447C22#571711yescharacteristic
448C4—42no—
449C4—44no—
450C4—62no—
451C4—82no—
452C4—122no—
453C4—121no—
454C4—122no—
455C4—121no—
456C4C2×#67⋊C411yescharacteristic
457C4C4×D3×D411yescharacteristic
458C4#561111yescharacteristic
459C4#567711yescharacteristic
460C3#2304511yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—122no—
464C2#7643311yescharacteristic
465C2#8180811yescharacteristic
466C2#8183511yescharacteristic
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1969C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1969D3
hypocenterC3#23045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1969

Complements

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

NH ≅ G/Ntype
2. #83874C2semidirect
5. C3×#1969C2semidirect
6. #70625C2semidirect
7. #70263C2semidirect
9. #69779C2semidirect
11. #69672C2semidirect
24. #5192C4semidirect
51. #1969D3semidirect
63. C2×C3×Q16C2×C4semidirect
97. #836C2×C4semidirect
102. #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
C2×#51922
#838742
#791462
#787742
C3×#19692
#706252
#702632
#697892
#697792
#696932
#696722
#230453