Finite Group Databasebeta

← #10818117

Subgroups — #10818117

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5160C211yesmaximal, characteristic
2#83823C211yesmaximal, characteristic, complemented
3#79116C211yesmaximal, characteristic, complemented
4#78726C212yesmaximal, complemented
5C3×#1972C211yesmaximal, characteristic, complemented
6#70572C211yesmaximal, characteristic, complemented
7#70195C212yesmaximal, complemented
8#69839C211yesmaximal, characteristic, complemented
9#69829C211yesmaximal, characteristic, complemented
10#69663C212yesmaximal, complemented
11#69642C212yesmaximal, complemented
12#23048—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14#5798C2212yes—
15C2×#932C2212yes—
16C2×#901C2211yescharacteristic
17#5504C2212yes—
18C2×C3×#106C2211yescharacteristic
19C2×C3×#102C2212yes—
20C2×C3×#97C2211yescharacteristic
21C3×#390C2212yes—
22C2×#833C2212yes—
23#5215C2212yes—
24#5160C418yescomplemented
25#5152—21no—
26#5152—21no—
27Dic3×D8—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#4785—22no—
40#4758—22no—
41#4757—22no—
42#4756—22no—
43#4552C2211yescharacteristic
44#4530C2212yes—
45#4523C2211yescharacteristic
46#4523C2212yes—
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2912—31no—
51#1972D311yescharacteristic, complemented
52#1951—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1773—31no—
57#1765—31no—
58C22×C3×D4C2312yes—
59C2×#235C2311yescharacteristic
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8—22no—
63C2×C3×D8C2×C414yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—42no—
68C2×C3×C42C2311yescharacteristic
69C2×#217—42no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74C2×#214—22no—
75#932—24no—
76#932C2×C418yescomplemented
77C2×#213—22no—
78C4×#67—42no—
79#926C2312yes—
80#926C2311yescharacteristic
81C2×#208C2311yescharacteristic
82C2×#208C2311yescharacteristic
83C2×#208C2312yes—
84#901—22no—
85#901C2×C414yescomplemented
86C2×ΓL(1,𝔽25)—21no—
87C2×D3×C8—21no—
88#893—22no—
89#892—22no—
90D3×C4⋊C4—22no—
91#887—22no—
92C3×#106—21no—
93C3×#106—21no—
94C3×#102—22no—
95C3×#102—22no—
96C3×#97—22no—
97#833C2×C418yescomplemented
98#832—22no—
99#832—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×AGL(1,ℤ/8)—32no—
116C22×D8C2×D311yescharacteristic
117C2×#131—31no—
118#520—32no—
119#518—32no—
120#472—38no—
121#442—62no—
122#442—61no—
123#442—61no—
124#440—62no—
125C4×SD16—62no—
126C4×D8—62no—
127C4×D8—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#418—32no—
138#416—31no—
139C2×#102C2×D312yes—
140C2×#100—32no—
141#405—32no—
142#405—31no—
143C2×#97C2×D311yescharacteristic
144#390C2×D312yes—
145#343—32no—
146#341—31no—
147#339—31no—
148#339—32no—
149C24×C3—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4C22×C414yes—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4C2411yescharacteristic
159C2×#67—21no—
160C22×Dic3—42no—
161C22×Dic3—42no—
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×D8—24no—
171C3×D8—24no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8C22×C412yes—
175C2×C3×C8C2×D412yes—
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8#4112yes—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C22⋊C4—84no—
182C3×C42—21no—
183C3×C42—21no—
184#217—22no—
185#217—21no—
186#217—42no—
187#217—42no—
188#214—28no—
189#213—28no—
190(C2×D3)⋊C4—42no—
191C4⋊Dic3C2×D412yes—
192C4⋊Dic3C2×D412yes—
193C4⋊Dic3C2×D412yes—
194Dic3⋊C4—42no—
195C4×Dic3—22no—
196C4×Dic3—21no—
197C4×Dic3—42no—
198#208—21no—
199#208—22no—
200#208—21no—
201#208—21no—
202#208C22×C414yes—
203#208C22×C412yes—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—21no—
207ΓL(1,𝔽25)—24no—
208D3×C8—24no—
209C2×#41—31no—
210C22×D4C22×D312yes—
211AGL(1,ℤ/8)—38no—
212AGL(1,ℤ/8)—64no—
213C2×SD16—62no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—22no—
217C2×D8—62no—
218C2×D8C4×D314yescomplemented
219#131—34no—
220#131—62no—
221C2×M16—31no—
222C2×M16—32no—
223C2×M16—31no—
224C2×M16—61no—
225C22×C8—61no—
226C22×C8C22×D311yescharacteristic
227C22×C8C22×D311yescharacteristic
228C4×Q8—62no—
229C4×D4—62no—
230C4×D4—62no—
231C4×D4—62no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—122no—
235#117—31no—
236#117—32no—
237#117—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4C22×D312yes—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—122no—
242C2×C42C22×D311yescharacteristic
243#108—34no—
244#107—32no—
245#107—62no—
246#107—62no—
247#106—21no—
248#106—21no—
249#106—31no—
250#106—31no—
251#106—62no—
252#106—62no—
253#104—62no—
254#103—32no—
255#102—22no—
256#102—22no—
257#102—32no—
258#102—62no—
259#102—62no—
260#100—38no—
261#97—22no—
262#97—62no—
263#97—61no—
264C4×C8—61no—
265C4×C8—62no—
266C23×C3—24no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—82no—
270C23×C3—82no—
271C23×C3H(5,𝔽2)11yescharacteristic
272C22×D3—21no—
273C3×D4—24no—
274C3×D4—24no—
275C3×D4—42no—
276C3×D4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—82no—
281C2×C3×C4C23×C411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287#67—22no—
288#67—22no—
289C2×Dic3—21no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—162no—
294C2×Dic3—162no—
295D12—21no—
296D12C4×D412yes—
297C4×D3—22no—
298C4×D3—22no—
299C4×D3—162no—
300Dic6—21no—
301Dic6C4×D412yes—
302C3×C8—22no—
303C3×C8C4×D412yes—
304C3×C8C4×D412yes—
305C3⋊C8—22no—
306C3⋊C8—24no—
307C3⋊C8—41no—
308C24—22no—
309#41—32no—
310#41—32no—
311#41—61no—
312#41—61no—
313C2×Q8—31no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—31no—
320C2×D4—61no—
321C2×D4C2×C4×D314yes—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—61no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C23×D311yescharacteristic
329SD16—68no—
330D8—24no—
331D8—24no—
332D8—68no—
333M16—32no—
334M16—34no—
335M16—61no—
336M16—62no—
337M16—64no—
338M16—61no—
339M16—61no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—62no—
343C2×C8—61no—
344C2×C8—64no—
345C2×C8—61no—
346C2×C8C2×C4×D312yes—
347C2×C8D3×D412yes—
348C2×C8D3×D411yescharacteristic
349C2×C8D3×D411yescharacteristic
350C2×C8#23712yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—32no—
354C4⋊C4—32no—
355C4⋊C4—32no—
356C4⋊C4—122no—
357C22⋊C4—61no—
358C22⋊C4—62no—
359C22⋊C4—84no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C42—21no—
364C42—21no—
365C42—61no—
366C42—62no—
367C42—122no—
368C22×C3—22no—
369C22×C3—44no—
370C22×C3—84no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—162no—
374C22×C3#52011yescharacteristic
375C22×C3D4211yescharacteristic
376C22×C3#55311yescharacteristic
377C2×D3—22no—
378C2×D3—42no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4C2×C4×D411yescharacteristic
382C3×C4C2×C4×D411yescharacteristic
383C3×C4#54711yescharacteristic
384C3×C4#54811yescharacteristic
385Dic3—22no—
386Dic3—42no—
387Dic3—42no—
388Dic3—42no—
389C23—24no—
390C23—42no—
391C23—42no—
392C23—61no—
393C23—82no—
394C23—82no—
395C23#100411yescharacteristic
396Q8—32no—
397Q8—61no—
398D4—24no—
399D4—24no—
400D4—32no—
401D4—42no—
402D4—42no—
403D4—62no—
404D4—62no—
405D4—61no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—82no—
416C2×C4—482no—
417C2×C4—482no—
418C2×C4—482no—
419C2×C4C22×C4×D311yescharacteristic
420C2×C4C2×D3×D411yescharacteristic
421C2×C4C2×D3×D411yescharacteristic
422C2×C4C2×#23711yescharacteristic
423C2×C4D3×#4111yescharacteristic
424C2×C4#101011yescharacteristic
425C8—22no—
426C8—62no—
427C8—64no—
428C8—121no—
429C8#67⋊C412yes—
430C8#67⋊C412yes—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—84no—
434C2×C3#236111yescharacteristic
435C2×C3#328812yes—
436D3—42no—
437C22—22no—
438C22—44no—
439C22—62no—
440C22—84no—
441C22—82no—
442C22—82no—
443C22—122no—
444C22—162no—
445C22#554611yescharacteristic
446C22#561411yescharacteristic
447C22#571711yescharacteristic
448C4—62no—
449C4—81no—
450C4—82no—
451C4—122no—
452C4—122no—
453C4—122no—
454C4C2×#67⋊C411yescharacteristic
455C4C4×D3×D411yescharacteristic
456C4#561111yescharacteristic
457C4#567711yescharacteristic
458C3#2304811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—84no—
462C2—122no—
463C2#7643311yescharacteristic
464C2#8186012yes—
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1972C2
socleC22×C3D42
prime radicalC2×#5160C2
cosocleC22×C3×C4C24
hypercenter#1972D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1972

Complements

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

NH ≅ G/Ntype
2. #83823C2semidirect
3. #79116C2semidirect
4. #78726C2semidirect
5. C3×#1972C2semidirect
6. #70572C2semidirect
7. #70195C2semidirect
8. #69839C2semidirect
9. #69829C2semidirect
10. #69663C2semidirect
11. #69642C2semidirect
24. #5160C4semidirect
51. #1972D3semidirect
63. C2×C3×D8C2×C4semidirect
76. #932C2×C4semidirect
85. #901C2×C4semidirect
97. #833C2×C4semidirect
101. #819C2×C4semidirect
218. C2×D8C4×D3semidirect
458. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C2×#51602
#838232
#791162
#787262
C3×#19722
#705722
#701952
#698392
#698292
#696632
#696422
#230483