Finite Group Databasebeta

← #10802092

Subgroups — #10802092

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5160C211yesmaximal, characteristic
2C2×#5152C211yesmaximal, characteristic, complemented
3#78914C212yesmaximal, complemented
4#78550C211yesmaximal, characteristic, complemented
5C3×#1978C211yesmaximal, characteristic, complemented
6#70720C211yesmaximal, characteristic, complemented
7#70692C211yesmaximal, characteristic, complemented
8#70318C212yesmaximal, complemented
9#70088C212yesmaximal, complemented
10Dic3×#100C212yesmaximal, complemented
11#69893C211yesmaximal, characteristic, complemented
12#21555—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×D4×Dic3C2212yes—
15C2×#932C2212yes—
16C2×#907C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5522C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#418C2212yes—
21C2×C3×#100C2212yes—
22C3×#405C2211yescharacteristic
23C2×#833C2212yes—
24C2×#832C2212yes—
25#5160C418yescomplemented
26#5152—21no—
27#5152—21no—
28#5152—21no—
29#5152—21no—
30#5133C2212yes—
31Dic3×M16C2212yes—
32Dic3×M16C2211yescharacteristic
33C2×#819C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#815C2211yescharacteristic
36#4997C2211yescharacteristic
37#4995C2212yes—
38#4785—22no—
39#4758—22no—
40#4757—22no—
41#4756—22no—
42#4708—21no—
43D3×#97—21no—
44#4689—21no—
45C4×ΓL(1,𝔽25)—21no—
46#4558C2212yes—
47#4554C2212yes—
48#4554C2212yes—
49C2×#472—31no—
50C4×AGL(1,ℤ/8)—32no—
51C2×#442—31no—
52#2875—31no—
53#1978D311yescharacteristic, complemented
54#1951—31no—
55#1931—32no—
56#1907—32no—
57#1815—31no—
58#1778—31no—
59C4×#100—32no—
60C22×C3×D4C2312yes—
61C23×Dic3—22no—
62C2×#235C2311yescharacteristic
63C2×C3×D8—21no—
64C2×C3×D8—21no—
65C2×C3×D8—22no—
66C2×C3×D8C2×C414yescomplemented
67C2×C3×M16C2312yes—
68C2×C3×M16C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#117C2311yescharacteristic
71C2×#217—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74D4×Dic3—22no—
75D4×Dic3—22no—
76C2×#214—22no—
77#932—24no—
78#932C2×C418yescomplemented
79C2×#213—22no—
80C2×C4⋊Dic3C2312yes—
81C2×C4×Dic3C2311yescharacteristic
82C2×#208C2311yescharacteristic
83C2×#208C2312yes—
84C22×C3⋊C8C2311yescharacteristic
85#907—22no—
86#907—22no—
87#901—22no—
88#901C2×C414yescomplemented
89C2×ΓL(1,𝔽25)—21no—
90C2×ΓL(1,𝔽25)—21no—
91C2×D3×C8—21no—
92C2×D3×C8—21no—
93#893D412yes—
94#887D412yes—
95#67⋊C4—22no—
96#872—21no—
97C42×D3—21no—
98C3×#108—22no—
99C3×#102—22no—
100C3×#102—22no—
101C3×#100—24no—
102C3×#97—21no—
103C3×C4×C8—21no—
104#833C2×C418yescomplemented
105#832—22no—
106#832—22no—
107#821—22no—
108#819C2×C414yescomplemented
109#817—21no—
110#817—21no—
111#816—22no—
112#815—21no—
113#815—22no—
114#815D411yescharacteristic
115#815D411yescharacteristic
116#815D411yescharacteristic
117#815D411yescharacteristic
118C8×Dic3—21no—
119C8×Dic3—22no—
120#808—22no—
121#807—22no—
122#803—21no—
123C4×C3⋊C8—21no—
124C2×AGL(1,ℤ/8)—32no—
125C22×D8C2×D311yescharacteristic
126C2×#131—31no—
127C2×#131—31no—
128C4×#41—31no—
129C2×C4×D4—32no—
130#472—38no—
131#442—61no—
132#442—61no—
133#442—61no—
134#442—61no—
135#442—62no—
136#440—62no—
137C4×SD16—62no—
138C4×D8—62no—
139C2×#108—31no—
140C2×#108C2×D311yescharacteristic
141#428—32no—
142C2×#106—31no—
143#418C2×D312yes—
144#417—32no—
145C2×#102—32no—
146C2×#100—32no—
147C2×#100C2×D312yes—
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#343—32no—
159#341—32no—
160C24×C3—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4C22×C414yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—21no—
169C22×Dic3—22no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C22×Dic3—84no—
173C22×Dic3#4112yes—
174#235—21no—
175#235—21no—
176#235C22×C412yes—
177#235#4112yes—
178C2×D12C2×D411yescharacteristic
179C2×C4×D3—21no—
180C2×C4×D3—22no—
181C2×Dic6C2×D411yescharacteristic
182C3×D8—24no—
183C3×D8—24no—
184C3×M16—21no—
185C3×M16—22no—
186C3×M16—22no—
187C3×M16#4112yes—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8C22×C412yes—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8#4112yes—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—21no—
196C3×C42—21no—
197#217—42no—
198#217—42no—
199#214—28no—
200#213—28no—
201(C2×D3)⋊C4—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204Dic3⋊C4—22no—
205C4×Dic3—22no—
206C4×Dic3C2×D412yes—
207C4×Dic3C2×D411yescharacteristic
208C4×Dic3C2×D411yescharacteristic
209#208—21no—
210#208—22no—
211#208C22×C414yes—
212#208C22×C412yes—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C2×D411yescharacteristic
218C2×C3⋊C8C2×D411yescharacteristic
219ΓL(1,𝔽25)—24no—
220ΓL(1,𝔽25)—24no—
221D3×C8—24no—
222D3×C8—24no—
223C2×#41—31no—
224C22×D4C22×D312yes—
225C23×C4—62no—
226AGL(1,ℤ/8)—38no—
227AGL(1,ℤ/8)—64no—
228C2×SD16—62no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—22no—
232C2×D8—62no—
233C2×D8C4×D314yescomplemented
234#131—34no—
235#131—62no—
236#131—62no—
237#131—62no—
238C2×M16—31no—
239C2×M16—32no—
240C2×M16—61no—
241C2×M16—61no—
242C2×M16C22×D312yes—
243C2×M16C22×D311yescharacteristic
244C22×C8—31no—
245C22×C8—61no—
246C22×C8—61no—
247C22×C8C22×D311yescharacteristic
248C4×Q8—32no—
249C4×D4—32no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—61no—
256#117C22×D311yescharacteristic
257C2×C4⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C42—31no—
260C2×C42—61no—
261#108—22no—
262#108—34no—
263#107—62no—
264#106—61no—
265#106—61no—
266#106—62no—
267#103—62no—
268#102—22no—
269#102—22no—
270#102—62no—
271#102—62no—
272#102—62no—
273#100—24no—
274#100—38no—
275#97—21no—
276#97—31no—
277#97—31no—
278#97—31no—
279#97—31no—
280#97—61no—
281#97—61no—
282#97—62no—
283C4×C8—21no—
284C4×C8—61no—
285C4×C8—61no—
286C4×C8—62no—
287C23×C3—24no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—82no—
291C23×C3—82no—
292C23×C3C2×#4111yescharacteristic
293C22×D3—21no—
294C3×D4—24no—
295C3×D4—24no—
296C3×D4—42no—
297C3×D4—42no—
298C2×C3×C4—22no—
299C2×C3×C4C23×C411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305#67—22no—
306#67—22no—
307C2×Dic3—21no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—84no—
311C2×Dic3—84no—
312C2×Dic3—82no—
313C2×Dic3#12112yes—
314C2×Dic3#12112yes—
315D12—21no—
316D12C4×D412yes—
317C4×D3—22no—
318C4×D3—22no—
319C4×D3—82no—
320Dic6—21no—
321Dic6C4×D412yes—
322C3×C8—22no—
323C3×C8—42no—
324C3×C8C4×D412yes—
325C3×C8C4×D412yes—
326C3⋊C8—22no—
327C3⋊C8—24no—
328C3⋊C8—21no—
329C3⋊C8—21no—
330C24—22no—
331#41—32no—
332#41—32no—
333#41—61no—
334#41—61no—
335C2×Q8—31no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—31no—
342C2×D4—61no—
343C2×D4C2×C4×D314yes—
344C22×C4—32no—
345C22×C4—62no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—244no—
351C22×C4C23×D311yescharacteristic
352SD16—68no—
353D8—24no—
354D8—24no—
355D8—68no—
356M16—21no—
357M16—22no—
358M16—22no—
359M16—32no—
360M16—34no—
361M16—64no—
362M16—61no—
363M16—62no—
364M16—64no—
365M16#23712yes—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—31no—
369C2×C8—31no—
370C2×C8—64no—
371C2×C8—61no—
372C2×C8—62no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—64no—
376C2×C8C2×C4×D312yes—
377C2×C8D3×D411yescharacteristic
378C2×C8D3×D411yescharacteristic
379C2×C8#23712yes—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4D3×D412yes—
384C22⋊C4—21no—
385C22⋊C4—62no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—21no—
389C42—31no—
390C42—32no—
391C42—31no—
392C42—62no—
393C22×C3—22no—
394C22×C3—44no—
395C22×C3—84no—
396C22×C3—84no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3C4×#4111yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#52911yescharacteristic
402C2×D3—22no—
403C2×D3—42no—
404C3×C4—42no—
405C3×C4C2×C4×D411yescharacteristic
406C3×C4C2×C4×D411yescharacteristic
407C3×C4#52511yescharacteristic
408C3×C4#53211yescharacteristic
409Dic3—22no—
410Dic3—22no—
411Dic3—22no—
412Dic3—82no—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—61no—
417C23—82no—
418C23—82no—
419C23D3×#4111yescharacteristic
420Q8—32no—
421Q8—61no—
422D4—24no—
423D4—24no—
424D4—32no—
425D4—42no—
426D4—42no—
427D4—62no—
428D4—62no—
429D4—61no—
430C2×C4—22no—
431C2×C4—32no—
432C2×C4—32no—
433C2×C4—61no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—242no—
439C2×C4—244no—
440C2×C4—244no—
441C2×C4—242no—
442C2×C4C22×C4×D311yescharacteristic
443C2×C4C2×D3×D411yescharacteristic
444C2×C4C2×D3×D411yescharacteristic
445C2×C4C2×#23711yescharacteristic
446C2×C4C2×#23711yescharacteristic
447C2×C4D3×#4111yescharacteristic
448C8—22no—
449C8—42no—
450C8—62no—
451C8—64no—
452C8—61no—
453C8—61no—
454C8#67⋊C412yes—
455C8#67⋊C412yes—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—84no—
459C2×C3C4×#12111yescharacteristic
460C2×C3#306812yes—
461D3—42no—
462C22—22no—
463C22—44no—
464C22—62no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—82no—
469C22—122no—
470C22C4×#23711yescharacteristic
471C22D3×#12111yescharacteristic
472C22#568011yescharacteristic
473C4—42no—
474C4—62no—
475C4—62no—
476C4—62no—
477C4—242no—
478C4C2×#67⋊C411yescharacteristic
479C4C4×D3×D411yescharacteristic
480C4#560611yescharacteristic
481C4#571611yescharacteristic
482C3#2155511yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—84no—
486C2—122no—
487C2#7629111yescharacteristic
488C2#8093812yes—
subgroupG/H
centerC22D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1978C2
socleC22×C3C2×#121
prime radicalC2×#5160C2
cosocleC22×C3×C4C24
hypercenter#1978D3
hypocenterC3#21555
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1978

Complements

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

NH ≅ G/Ntype
2. C2×#5152C2semidirect
3. #78914C2semidirect
4. #78550C2semidirect
5. C3×#1978C2semidirect
6. #70720C2semidirect
7. #70692C2semidirect
8. #70318C2semidirect
9. #70088C2semidirect
10. Dic3×#100C2semidirect
11. #69893C2semidirect
25. #5160C4semidirect
53. #1978D3semidirect
66. C2×C3×D8C2×C4semidirect
78. #932C2×C4semidirect
88. #901C2×C4semidirect
104. #833C2×C4semidirect
108. #819C2×C4semidirect
233. C2×D8C4×D3semidirect
482. C3#21555semidirect

Maximal subgroups

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

subgroupindex
C2×#51602
C2×#51522
#789142
#785502
C3×#19782
#707202
#706922
#703182
#700882
Dic3×#1002
#698932
#215553