Finite Group Databasebeta

← #10802095

Subgroups — #10802095

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4896C211yesmaximal, characteristic
2#79200C212yesmaximal
3C2×#4853C211yesmaximal, characteristic, complemented
4#78550C211yesmaximal, characteristic
5C3×#1951C211yesmaximal, characteristic, complemented
6#70692C211yesmaximal, characteristic, complemented
7#70672C212yesmaximal, complemented
8#70592C211yesmaximal, characteristic
9#69904C211yesmaximal, characteristic, complemented
10#69007C212yesmaximal
11#68973C212yesmaximal, complemented
12#21556—31nomaximal, Sylow
13C2×#909C2212yes—
14C2×#907C2211yescharacteristic
15C22×Dic12C2211yescharacteristic
16C2×#901C2211yescharacteristic
17#5522C2211yescharacteristic
18C2×#887C2212yes—
19C2×C3×#108C2211yescharacteristic
20C3×#428C2212yes—
21C2×C3×#106C2211yescharacteristic
22C3×#405C2211yescharacteristic
23C2×#824C2212yes—
24C2×#824C2212yes—
25Dic3×M16C2212yes—
26Dic3×M16C2211yescharacteristic
27C2×#819C2211yescharacteristic
28C2×#815C2211yescharacteristic
29#5004C2212yes—
30#4997C2211yescharacteristic
31C2×#810C2212yes—
32#4966C2212yes—
33#4896C418yescomplemented
34#4889—22no—
35#4873—22no—
36#4853—22no—
37#4853—21no—
38#4853—21no—
39#4853—21no—
40#4853—21no—
41#4852—22no—
42#4708—21no—
43D3×#97—21no—
44#4689—21no—
45C4×ΓL(1,𝔽25)—21no—
46C3×#343C2212yes—
47#4558C2212yes—
48#4530C2212yes—
49C2×#473—31no—
50C4×#137—32no—
51C2×#441—31no—
52#2875—31no—
53#1979—31no—
54#1951D311yescharacteristic, complemented
55#1930—32no—
56#1908—32no—
57#1815—31no—
58#1778—31no—
59C4×#101—32no—
60C2×#235C2311yescharacteristic
61C22×Dic6C2312yes—
62C2×C3×M16C2312yes—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#117C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×Dic3⋊C4—22no—
68C2×C4×Dic3—22no—
69C2×C4×Dic3C2311yescharacteristic
70C2×#208C2312yes—
71C2×#208C2311yescharacteristic
72C22×C3⋊C8C2311yescharacteristic
73#909—24no—
74#909C2×C418yes—
75#907—22no—
76#907—22no—
77C2×Dic12—22no—
78C2×Dic12—21no—
79C2×Dic12—21no—
80C2×Dic12—22no—
81C2×Dic12C2×C414yescomplemented
82C2×#204—22no—
83#901—22no—
84#901C2×C414yes—
85C2×ΓL(1,𝔽25)—21no—
86C2×ΓL(1,𝔽25)—21no—
87C2×D3×C8—21no—
88C2×D3×C8—21no—
89#893D412yes—
90#887—22no—
91#887—22no—
92#887—22no—
93#887—22no—
94#887D412yes—
95#67⋊C4—22no—
96#872—21no—
97C42×D3—21no—
98C3×#108C2×C414yescomplemented
99C3×#107—22no—
100C3×#106—21no—
101C3×#106—21no—
102C3×#106—22no—
103C3×#97—21no—
104C3×C4×C8—21no—
105#824—24no—
106#824C2×C418yescomplemented
107#819—22no—
108#815—22no—
109#815—21no—
110#815D411yescharacteristic
111#815D411yescharacteristic
112#815D411yescharacteristic
113#815D411yescharacteristic
114C8×Dic3—22no—
115C8×Dic3—21no—
116#810—22no—
117#810—22no—
118#810—22no—
119#810—22no—
120#810—22no—
121#809—22no—
122#803—21no—
123C4×C3⋊C8—21no—
124C2×#137—32no—
125C22×Q16—31no—
126C2×#131—31no—
127C2×#131—31no—
128C4×#41—31no—
129C2×C4×Q8—32no—
130#473—38no—
131#441—61no—
132#441—61no—
133#441—61no—
134#441—61no—
135#441—62no—
136#440—62no—
137C4×Q16—62no—
138C4×SD16—62no—
139C2×#108—31no—
140C2×#108C2×D311yescharacteristic
141#428C2×D312yes—
142C2×#106C2×D311yescharacteristic
143#419—32no—
144#417—32no—
145C2×#103—32no—
146C2×#101—32no—
147C2×#101—32no—
148#405—31no—
149#405—61no—
150#405—61no—
151#405C2×D311yescharacteristic
152C4×M16—31no—
153C4×M16—32no—
154C4×M16—61no—
155C2×#97—31no—
156C2×#97—61no—
157#343—32no—
158#343C2×D312yes—
159#341—32no—
160C22×C3×C4—22no—
161C22×C3×C4C2411yescharacteristic
162C2×#67—21no—
163C22×Dic3—22no—
164C22×Dic3#4112yes—
165#235—21no—
166#235—21no—
167#235C22×C412yes—
168#235#4112yes—
169C2×D12C2×D411yescharacteristic
170C2×C4×D3—21no—
171C2×C4×D3—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6C22×C414yes—
178C2×Dic6C2×D411yescharacteristic
179C3×M16—21no—
180C3×M16—22no—
181C3×M16C22×C414yes—
182C3×M16#4112yes—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8C22×C412yes—
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8C2×D411yescharacteristic
188C2×C3×C8#4112yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—21no—
193C3×C42—21no—
194(C2×D3)⋊C4—22no—
195Dic3⋊C4—22no—
196Dic3⋊C4—42no—
197Dic3⋊C4—42no—
198C4×Dic3—22no—
199C4×Dic3—42no—
200C4×Dic3—42no—
201C4×Dic3C2×D412yes—
202C4×Dic3C2×D411yescharacteristic
203C4×Dic3C2×D411yescharacteristic
204#208—21no—
205#208—22no—
206#208—22no—
207#208C22×C412yes—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8C2×D411yescharacteristic
213C2×C3⋊C8C2×D411yescharacteristic
214Dic12—28no—
215Dic12—24no—
216Dic12—24no—
217#204—28no—
218ΓL(1,𝔽25)—24no—
219ΓL(1,𝔽25)—24no—
220D3×C8—24no—
221D3×C8—24no—
222C2×#41—31no—
223C22×Q8—32no—
224#137—38no—
225#137—64no—
226C2×Q16—34no—
227C2×Q16—62no—
228C2×Q16—61no—
229C2×Q16—61no—
230C2×Q16—62no—
231C2×SD16—62no—
232#131—34no—
233#131—62no—
234#131—62no—
235#131—62no—
236C2×M16—31no—
237C2×M16—32no—
238C2×M16—61no—
239C2×M16—61no—
240C2×M16C22×D312yes—
241C2×M16C22×D311yescharacteristic
242C22×C8—31no—
243C22×C8—61no—
244C22×C8—61no—
245C22×C8C22×D311yescharacteristic
246C4×Q8—32no—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—32no—
252C4×D4—62no—
253#117—61no—
254#117C22×D311yescharacteristic
255C2×C4⋊C4—62no—
256C2×C4⋊C4C22×D312yes—
257C2×C42—31no—
258C2×C42—61no—
259C2×C42—62no—
260#108—62no—
261#108C4×D314yescomplemented
262#107—22no—
263#106—21no—
264#106—21no—
265#106—22no—
266#103—62no—
267#103—62no—
268#103—62no—
269#103—62no—
270#103—62no—
271#102—62no—
272#101—38no—
273#101—64no—
274#97—21no—
275#97—31no—
276#97—31no—
277#97—31no—
278#97—31no—
279#97—61no—
280#97—61no—
281#97—62no—
282C4×C8—21no—
283C4×C8—61no—
284C4×C8—61no—
285C4×C8—62no—
286C23×C3C2×#4111yescharacteristic
287C22×D3—21no—
288C2×C3×C4—22no—
289C2×C3×C4—42no—
290C2×C3×C4—82no—
291C2×C3×C4C23×C411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297#67—22no—
298#67—22no—
299C2×Dic3—21no—
300C2×Dic3—24no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3#12112yes—
305C2×Dic3#12112yes—
306D12—21no—
307D12C4×D412yes—
308C4×D3—22no—
309C4×D3—22no—
310C4×D3—82no—
311Dic6—21no—
312Dic6—24no—
313Dic6—24no—
314Dic6—42no—
315Dic6—42no—
316Dic6C4×D412yes—
317C3×C8—24no—
318C3×C8—22no—
319C3×C8C4×D412yes—
320C3×C8C4×D412yes—
321C3⋊C8—22no—
322C3⋊C8—21no—
323C3⋊C8—21no—
324C3⋊C8—42no—
325#41—32no—
326#41—32no—
327#41—61no—
328#41—61no—
329C2×Q8—34no—
330C2×Q8—31no—
331C2×Q8—62no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×D4—31no—
337C2×D4—61no—
338C22×C4—22no—
339C22×C4—32no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4C23×D311yescharacteristic
344Q16—68no—
345Q16—64no—
346Q16—64no—
347SD16—68no—
348M16—21no—
349M16—22no—
350M16—32no—
351M16—64no—
352M16—61no—
353M16—64no—
354M16—62no—
355M16—62no—
356M16C2×C4×D314yes—
357M16#23912yes—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—31no—
361C2×C8—31no—
362C2×C8—64no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—64no—
367C2×C8—62no—
368C2×C8C2×C4×D312yes—
369C2×C8D3×D411yescharacteristic
370C2×C8D3×D411yescharacteristic
371C2×C8#23912yes—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—62no—
375C4⋊C4—122no—
376C4⋊C4—122no—
377C4⋊C4D3×D412yes—
378C22⋊C4—21no—
379C22⋊C4—62no—
380C42—21no—
381C42—31no—
382C42—32no—
383C42—31no—
384C42—62no—
385C42—122no—
386C42—122no—
387C22×C3—22no—
388C22×C3C4×#4111yescharacteristic
389C22×C3C2×#12111yescharacteristic
390C22×C3#52911yescharacteristic
391C2×D3—22no—
392C2×D3—42no—
393C3×C4—42no—
394C3×C4—82no—
395C3×C4C2×C4×D411yescharacteristic
396C3×C4C2×C4×D411yescharacteristic
397C3×C4#52511yescharacteristic
398C3×C4#53211yescharacteristic
399Dic3—22no—
400Dic3—22no—
401Dic3—22no—
402Dic3—44no—
403C23—61no—
404C23D3×#4111yescharacteristic
405Q8—32no—
406Q8—61no—
407Q8—64no—
408Q8—64no—
409Q8—122no—
410Q8—122no—
411D4—32no—
412D4—62no—
413D4—62no—
414D4—61no—
415C2×C4—22no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—42no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—64no—
423C2×C4—82no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—242no—
428C2×C4C22×C4×D311yescharacteristic
429C2×C4C2×D3×D411yescharacteristic
430C2×C4C2×D3×D411yescharacteristic
431C2×C4C2×#23911yescharacteristic
432C2×C4C2×#23911yescharacteristic
433C2×C4D3×#4111yescharacteristic
434C8—24no—
435C8—22no—
436C8—62no—
437C8—61no—
438C8—61no—
439C8—122no—
440C8#89312yes—
441C8#89312yes—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3C4×#12111yescharacteristic
445C2×C3#306912yes—
446D3—42no—
447C22—22no—
448C22—62no—
449C22—122no—
450C22C4×#23911yescharacteristic
451C22D3×#12111yescharacteristic
452C22#569111yescharacteristic
453C4—42no—
454C4—62no—
455C4—62no—
456C4—62no—
457C4—82no—
458C4—124no—
459C4C2×#89311yescharacteristic
460C4C4×D3×D411yescharacteristic
461C4#562711yescharacteristic
462C4#572511yescharacteristic
463C3#2155611yescharacteristic, complemented, Sylow
464C2—21no—
465C2—21no—
466C2—122no—
467C2#7596411yescharacteristic
468C2#8093112yes—
subgroupG/H
centerC22D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1951C2
socleC22×C3C2×#121
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1951D3
hypocenterC3#21556
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
3. C2×#4853C2semidirect
5. C3×#1951C2semidirect
6. #70692C2semidirect
7. #70672C2semidirect
9. #69904C2semidirect
11. #68973C2semidirect
33. #4896C4semidirect
54. #1951D3semidirect
81. C2×Dic12C2×C4semidirect
98. C3×#108C2×C4semidirect
106. #824C2×C4semidirect
261. #108C4×D3semidirect
463. C3#21556semidirect

Maximal subgroups

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

subgroupindex
C2×#48962
#792002
C2×#48532
#785502
C3×#19512
#706922
#706722
#705922
#699042
#690072
#689732
#215563