Finite Group Databasebeta

← #10818106

Subgroups — #10818106

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5192C211yesmaximal, characteristic
2#83874C211yesmaximal, characteristic, complemented
3#79116C211yesmaximal, characteristic
4#78774C212yesmaximal
5C3×#1971C211yesmaximal, characteristic, complemented
6#70572C211yesmaximal, characteristic, complemented
7#70263C212yesmaximal, complemented
8#69839C211yesmaximal, characteristic
9#69829C211yesmaximal, characteristic, complemented
10#69693C212yesmaximal
11#69672C212yesmaximal, complemented
12#23047—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14#5812C2212yes—
15C2×#942C2212yes—
16C2×#901C2211yescharacteristic
17#5504C2212yes—
18C2×C3×#106C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#97C2211yescharacteristic
21C3×#391C2212yes—
22C2×#836C2212yes—
23#5225C2212yes—
24#5192C418yescomplemented
25#5184—21no—
26#5184—21no—
27Dic3×Q16—22no—
28#5113C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#5106C2211yescharacteristic
31#5101C2212yes—
32#5101C2211yescharacteristic
33#4964C2212yes—
34#4964C2212yes—
35C2×#805C2211yescharacteristic
36#4861—22no—
37#4860—21no—
38D3×#106—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#2912—31no—
51#1971D311yescharacteristic, complemented
52#1951—31no—
53#1930—32no—
54#1882—32no—
55#1857—32no—
56#1773—31no—
57#1765—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×#106—21no—
92C3×#106—21no—
93C3×#103—22no—
94C3×#103—22no—
95C3×#97—22no—
96C3×#95—42no—
97#836C2×C418yescomplemented
98#835—22no—
99#835—22no—
100#821D412yes—
101#819C2×C414yescomplemented
102#818D412yes—
103#817D411yescharacteristic
104#817D411yescharacteristic
105#816D412yes—
106#815—21no—
107#815—22no—
108C8×Dic3—21no—
109C8×Dic3—22no—
110#808—22no—
111#808—22no—
112#807—22no—
113#807—22no—
114#805—22no—
115C2×#137—32no—
116C22×Q16C2×D311yescharacteristic
117C2×#131—31no—
118#520—32no—
119#519—32no—
120#473—38no—
121#441—62no—
122#441—61no—
123#441—61no—
124#440—62no—
125C4×Q16—62no—
126C4×Q16—62no—
127C4×SD16—62no—
128C2×#108—31no—
129#428—32no—
130#428—32no—
131#428—62no—
132#427—31no—
133#427—61no—
134C2×#106—61no—
135C2×#106C2×D311yescharacteristic
136C2×#104—31no—
137#419—32no—
138#416—31no—
139C2×#103C2×D312yes—
140C2×#101—32no—
141#405—32no—
142#405—31no—
143C2×#97C2×D311yescharacteristic
144#391C2×D312yes—
145#343—32no—
146#341—31no—
147#339—31no—
148#339—32no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C22×C414yes—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4C2411yescharacteristic
159C2×#67—21no—
160#235—21no—
161#235—21no—
162#235C22×C412yes—
163#235#4112yes—
164C2×D12C2×D411yescharacteristic
165C2×C4×D3—21no—
166C2×C4×D3—42no—
167C2×Dic6C2×D411yescharacteristic
168C3×Q16—24no—
169C3×Q16—24no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8C22×C412yes—
173C2×C3×C8C2×D412yes—
174C2×C3×C8C2×D411yescharacteristic
175C2×C3×C8C2×D411yescharacteristic
176C2×C3×C8#4112yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C42—21no—
180C3×C42—21no—
181#217—21no—
182#217—22no—
183#216—28no—
184#215—28no—
185(C2×D3)⋊C4—42no—
186C4⋊Dic3—42no—
187C4⋊Dic3—42no—
188C4⋊Dic3C2×D412yes—
189C4⋊Dic3C2×D412yes—
190C4⋊Dic3C2×D412yes—
191Dic3⋊C4—42no—
192C4×Dic3—22no—
193C4×Dic3—21no—
194C4×Dic3—42no—
195C4×Dic3—42no—
196C4×Dic3—42no—
197#208—21no—
198#208—22no—
199#208—21no—
200#208—21no—
201#208C22×C414yes—
202#208C22×C412yes—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—21no—
206ΓL(1,𝔽25)—24no—
207D3×C8—24no—
208C2×#41—31no—
209C22×Q8C22×D312yes—
210#137—38no—
211#137—64no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—22no—
215C2×Q16—62no—
216C2×Q16C4×D314yescomplemented
217C2×SD16—62no—
218#131—34no—
219#131—62no—
220C2×M16—31no—
221C2×M16—32no—
222C2×M16—31no—
223C2×M16—61no—
224C22×C8—61no—
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×Q8—62no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×Q8—62no—
232C4×D4—62no—
233C4×D4—122no—
234#117—31no—
235#117—32no—
236#117—62no—
237#117—122no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4C22×D312yes—
240C2×C42C22×D311yescharacteristic
241#108—34no—
242#107—32no—
243#107—62no—
244#107—62no—
245#106—21no—
246#106—21no—
247#106—31no—
248#106—31no—
249#106—62no—
250#106—62no—
251#104—62no—
252#103—22no—
253#103—22no—
254#103—32no—
255#103—62no—
256#103—62no—
257#102—32no—
258#101—38no—
259#97—22no—
260#97—62no—
261#97—61no—
262C4×C8—62no—
263C4×C8—61no—
264#95—42no—
265C23×C3H(5,𝔽2)11yescharacteristic
266C22×D3—21no—
267C3×Q8—24no—
268C3×Q8—24no—
269C3×Q8—42no—
270C3×Q8—42no—
271C2×C3×C4—24no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—82no—
278C2×C3×C4C23×C411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C42−1+411yescharacteristic
284#67—22no—
285#67—22no—
286C2×Dic3—21no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290D12—21no—
291D12C4×D412yes—
292C4×D3—22no—
293C4×D3—22no—
294C4×D3—162no—
295Dic6—21no—
296Dic6C4×D412yes—
297C3×C8—22no—
298C3×C8C4×D412yes—
299C3×C8C4×D412yes—
300C3⋊C8—22no—
301C3⋊C8—24no—
302C3⋊C8—41no—
303#41—32no—
304#41—32no—
305#41—61no—
306#41—61no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—31no—
313C2×Q8C2×C4×D314yes—
314C2×D4—31no—
315C2×D4—61no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—61no—
320C22×C4—122no—
321C22×C4C23×D311yescharacteristic
322Q16—24no—
323Q16—24no—
324Q16—68no—
325SD16—68no—
326M16—32no—
327M16—34no—
328M16—61no—
329M16—62no—
330M16—64no—
331M16—61no—
332M16—61no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—62no—
336C2×C8—61no—
337C2×C8—64no—
338C2×C8—61no—
339C2×C8C2×C4×D312yes—
340C2×C8D3×D412yes—
341C2×C8D3×D411yescharacteristic
342C2×C8D3×D411yescharacteristic
343C2×C8#23712yes—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—32no—
347C4⋊C4—32no—
348C4⋊C4—32no—
349C4⋊C4—122no—
350C4⋊C4—122no—
351C4⋊C4—122no—
352C22⋊C4—61no—
353C22⋊C4—62no—
354C22⋊C4—122no—
355C42—21no—
356C42—21no—
357C42—62no—
358C42—61no—
359C42—122no—
360C42—122no—
361C42—122no—
362C22×C3—22no—
363C22×C3#52011yescharacteristic
364C22×C3D4211yescharacteristic
365C22×C3#55311yescharacteristic
366C2×D3—22no—
367C2×D3—42no—
368C3×C4—44no—
369C3×C4—81no—
370C3×C4—82no—
371C3×C4C2×C4×D411yescharacteristic
372C3×C4C2×C4×D411yescharacteristic
373C3×C4#54711yescharacteristic
374C3×C4#54811yescharacteristic
375Dic3—22no—
376Dic3—42no—
377Dic3—42no—
378Dic3—42no—
379C23—61no—
380C23#100411yescharacteristic
381Q8—24no—
382Q8—24no—
383Q8—32no—
384Q8—42no—
385Q8—42no—
386Q8—61no—
387D4—32no—
388D4—62no—
389D4—62no—
390D4—61no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—61no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—82no—
404C2×C4—482no—
405C2×C4C22×C4×D311yescharacteristic
406C2×C4C2×D3×D411yescharacteristic
407C2×C4C2×D3×D411yescharacteristic
408C2×C4C2×#23711yescharacteristic
409C2×C4D3×#4111yescharacteristic
410C2×C4#101011yescharacteristic
411C8—22no—
412C8—62no—
413C8—64no—
414C8—121no—
415C8#67⋊C412yes—
416C8#67⋊C412yes—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3#236111yescharacteristic
420C2×C3#328912yes—
421D3—42no—
422C22—22no—
423C22—62no—
424C22—122no—
425C22#554611yescharacteristic
426C22#561411yescharacteristic
427C22#571711yescharacteristic
428C4—44no—
429C4—62no—
430C4—81no—
431C4—82no—
432C4—122no—
433C4—122no—
434C4—122no—
435C4C2×#67⋊C411yescharacteristic
436C4C4×D3×D411yescharacteristic
437C4#561111yescharacteristic
438C4#567711yescharacteristic
439C3#2304711yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—122no—
443C2#7643311yescharacteristic
444C2#8188512yes—
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1971C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1971D3
hypocenterC3#23047
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1971

Complements

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

NH ≅ G/Ntype
2. #83874C2semidirect
5. C3×#1971C2semidirect
6. #70572C2semidirect
7. #70263C2semidirect
9. #69829C2semidirect
11. #69672C2semidirect
24. #5192C4semidirect
51. #1971D3semidirect
63. C2×C3×Q16C2×C4semidirect
97. #836C2×C4semidirect
101. #819C2×C4semidirect
216. C2×Q16C4×D3semidirect
439. C3#23047semidirect

Maximal subgroups

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

subgroupindex
C2×#51922
#838742
#791162
#787742
C3×#19712
#705722
#702632
#698392
#698292
#696932
#696722
#230473