Finite Group Databasebeta

← #10798400

Subgroups — #10798400

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2276C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#73415C211yesmaximal, characteristic
4#71633C212yesmaximal
5#71631C211yesmaximal, characteristic, complemented
6#71628C212yesmaximal, complemented
7C4×#807C211yesmaximal, characteristic, complemented
8#69474C211yesmaximal, characteristic
9#69462C212yesmaximal
10#69459C212yesmaximal, complemented
11#69435C211yesmaximal, characteristic, complemented
12#21136—31nomaximal, Sylow
13C2×C3×C4×Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C22×#215C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×C4×C4⋊C4—21no—
18C3×C4×C4⋊C4—22no—
19C3×C4×C4⋊C4C2212yes—
20C3×C4×C4⋊C4C2211yescharacteristic
21C3×#376—21no—
22C3×#376C2211yescharacteristic
23C2×#835C2212yes—
24C2×#835C2211yescharacteristic
25C4×#215C418yescomplemented
26C2×#809C2212yes—
27C2×#809C2211yescharacteristic
28C2×#808C2211yescharacteristic
29C2×#807C2212yes—
30#4939C2212yes—
31C4×C4⋊Dic3C2211yescharacteristic
32#4928C2212yes—
33C2×#803C2211yescharacteristic
34C2×C4×C3⋊C8C2211yescharacteristic
35#4669—22no—
36#4667—21no—
37#4664—21no—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4524C2212yes—
42#4524C2212yes—
43C2×C4×SD16—31no—
44#2319—31no—
45#2276D311yescharacteristic, complemented
46#1925—32no—
47#1919—32no—
48#1880—32no—
49#1863—32no—
50C4×#107—31no—
51#1763—31no—
52#1762—31no—
53#1746—31no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C3×C4×Q8—24no—
58C3×C4×Q8—24no—
59C3×C4×Q8—22no—
60C3×C4×Q8C2×C414yes—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42—22no—
66C2×C3×C42—21no—
67C2×C3×C42—22no—
68C2×C3×C42—21no—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×#215—24no—
72C2×#215C2×C414yes—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—21no—
80C22×C3⋊C8C2312yes—
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C4×D12—22no—
84C4×D12C2×C414yes—
85C3×#95—22no—
86C3×#95—21no—
87C3×#95D411yescharacteristic
88C3×#95D411yescharacteristic
89C3×#95D412yes—
90#835—24no—
91#835C2×C414yes—
92C3⋊#95—21no—
93C3⋊#95—22no—
94#809—24no—
95#809C2×C414yes—
96#808C2×C414yes—
97#807—24no—
98#803—22no—
99C4×C3⋊C8C2×C414yes—
100C22×SD16—31no—
101C2×C4×Q8C2×D312yes—
102C2×C4×Q8C2×D311yescharacteristic
103C2×C4×D4—31no—
104C4×SD16—38no—
105C2×#107—32no—
106C2×#106—31no—
107C2×#103—31no—
108C2×#103—32no—
109C2×#102—31no—
110C2×#102—32no—
111C2×#97—31no—
112C2×C4×C8—31no—
113#390—32no—
114#389—32no—
115#388—62no—
116#381—61no—
117#380—61no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—31no—
121C4×C4⋊C4C2×D312yes—
122C4×C4⋊C4C2×D311yescharacteristic
123#376—21no—
124#376C2×D311yescharacteristic
125#340—32no—
126#340—32no—
127#340—32no—
128#336—32no—
129#336—32no—
130C23×D3—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—22no—
133C2×C3×Q8C22×C412yes—
134C2×C3×Q8#4114yes—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—22no—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12—21no—
148C2×D12C22×C412yes—
149C2×C4×D3—24no—
150C2×C4×D3—82no—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4C22×C412yes—
156C3×C4⋊C4#4114yes—
157C3×C42—28no—
158C3×C42—24no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—81no—
162C3×C42—82no—
163C3×C42C22×C412yes—
164C3×C42#4112yes—
165#215—28no—
166(C2×D3)⋊C4—24no—
167(C2×D3)⋊C4—82no—
168(C2×D3)⋊C4—82no—
169C4⋊Dic3—21no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3C22×C412yes—
173Dic3⋊C4—82no—
174C4×Dic3—81no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8C22×C412yes—
181C2×C3⋊C8C22×C412yes—
182C22×Q8C22×D311yescharacteristic
183C22×D4—31no—
184C23×C4—61no—
185C2×SD16—34no—
186C2×SD16—64no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8—32no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—22no—
193C4×Q8C4×D314yes—
194C4×D4—34no—
195C4×D4—62no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C22×D311yescharacteristic
202C2×C4⋊C4C22×D312yes—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—61no—
211C2×C42C22×D312yes—
212C2×C42C22×D311yescharacteristic
213#107—64no—
214#106—34no—
215#103—34no—
216#103—64no—
217#102—34no—
218#102—64no—
219#97—62no—
220C4×C8—34no—
221#95—22no—
222#95—21no—
223#95—61no—
224#95—62no—
225#95#6711yescharacteristic
226#95#6711yescharacteristic
227#95#6712yes—
228C23×C3C22×D411yescharacteristic
229C22×D3—22no—
230C22×D3—42no—
231C22×D3—41no—
232C22×D3—42no—
233C22×D3—41no—
234C3×Q8—22no—
235C3×Q8—24no—
236C3×Q8#11714yes—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4C4×D412yes—
255C2×C3×C4C4×D412yes—
256C2×C3×C4C23×C411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4C2×#4112yes—
259C2×C3×C4C2×#4111yescharacteristic
260C2×Dic3—22no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265D12—22no—
266D12—22no—
267D12—42no—
268C4×D3—84no—
269C3⋊C8—22no—
270C3⋊C8—22no—
271C3⋊C8—42no—
272C24—61no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8C2×C4×D312yes—
276C2×Q8#23514yes—
277C2×D4—32no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—61no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—61no—
285C22×C4—64no—
286C22×C4—62no—
287C22×C4—242no—
288C22×C4C2×#6712yes—
289C22×C4C2×#6712yes—
290C22×C4C2×#6711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C22×C4C23×D311yescharacteristic
293SD16—68no—
294C2×C8—32no—
295C2×C8—32no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—62no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—32no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—242no—
310C4⋊C4C2×C4×D312yes—
311C4⋊C4#23514yes—
312C22⋊C4—64no—
313C22⋊C4—242no—
314C22⋊C4—242no—
315C42—28no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—81no—
320C42—82no—
321C42—241no—
322C42C2×C4×D312yes—
323C42#23512yes—
324C22×C3C2×C4×D411yescharacteristic
325C22×C3#52512yes—
326C22×C3#57611yescharacteristic
327C22×C3#57712yes—
328C22×C3#57811yescharacteristic
329C2×D3—42no—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—82no—
333C2×D3—82no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—24no—
337C3×C4—42no—
338C3×C4—44no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4C2×#11711yescharacteristic
342C3×C4C4×#4111yescharacteristic
343C3×C4#52912yes—
344Dic3—42no—
345Dic3—82no—
346C23—62no—
347C23—122no—
348C23—121no—
349C23—122no—
350C23—121no—
351C23C22×#6711yescharacteristic
352Q8—22no—
353Q8—24no—
354Q8#87214yes—
355D4—62no—
356D4—62no—
357D4—122no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—62no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—244no—
381C2×C4C4×#6712yes—
382C2×C4C4×#6712yes—
383C2×C4C22×C4×D311yescharacteristic
384C2×C4C2×#23512yes—
385C2×C4C2×#23512yes—
386C2×C4C2×#23511yescharacteristic
387C8—62no—
388C8—62no—
389C8—122no—
390C2×C3#229011yescharacteristic
391C2×C3#294911yescharacteristic
392C2×C3#295111yescharacteristic
393C2×C3#311912yes—
394C2×C3#312112yes—
395D3—82no—
396C22—122no—
397C22—122no—
398C22—122no—
399C22—242no—
400C22—242no—
401C22C2×C4×#6711yescharacteristic
402C22#579112yes—
403C22#583511yescharacteristic
404C22#583712yes—
405C22#583811yescharacteristic
406C4—22no—
407C4—22no—
408C4—24no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—44no—
413C4—122no—
414C4—242no—
415C4C2×#87211yescharacteristic
416C4C4×#23511yescharacteristic
417C4#547912yes—
418C3#2113611yescharacteristic, complemented, Sylow
419C2—242no—
420C2#8206511yescharacteristic
421C2#8254811yescharacteristic
422C2#8255212yes—
423C2#8255711yescharacteristic
424C2#8256812yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2276C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#2276D3
hypocenterC3#21136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
1. C3×#2276C2semidirect
5. #71631C2semidirect
6. #71628C2semidirect
7. C4×#807C2semidirect
10. #69459C2semidirect
11. #69435C2semidirect
25. C4×#215C4semidirect
45. #2276D3semidirect
418. C3#21136semidirect

Maximal subgroups

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

subgroupindex
C3×#22762
C2×C4×#2152
#734152
#716332
#716312
#716282
C4×#8072
#694742
#694622
#694592
#694352
#211363