Finite Group Databasebeta

← #10818112

Subgroups — #10818112

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
1C3×#2912C211yesmaximal, characteristic, complemented
2C2×#5142C211yesmaximal, characteristic
3#82564C212yesmaximal, complemented
4#79167C211yesmaximal, characteristic, complemented
5#72009C211yesmaximal, characteristic, complemented
6#71667C212yesmaximal, complemented
7#70674C212yesmaximal, complemented
8#70673C212yesmaximal, complemented
9#70660C211yesmaximal, characteristic, complemented
10#70576C211yesmaximal, characteristic, complemented
11#70572C211yesmaximal, characteristic, complemented
12#23048—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14C3×#520C2212yes—
15C2×#949C2212yes—
16C22×D24C2211yescharacteristic
17#5521C2212yes—
18C3×#428—22no—
19C3×#428C2212yes—
20C3×#427—21no—
21C3×#427C2211yescharacteristic
22C2×C3×#106—21no—
23C2×C3×#106C2211yescharacteristic
24C2×#837C2211yescharacteristic
25#5233C2211yescharacteristic
26#5142C418yescomplemented
27C2×#823C2212yes—
28C2×#819C2211yescharacteristic
29#5101C2211yescharacteristic
30C2×#815C2211yescharacteristic
31#5026—22no—
32C4×#215—22no—
33#5014—22no—
34C4×#213—22no—
35#5001C2212yes—
36#4997C2212yes—
37#4988C2212yes—
38C2×#809C2212yes—
39#4964C2212yes—
40#4872—22no—
41#4869—21no—
42#4869—21no—
43#4558C2211yescharacteristic
44#4555C2211yescharacteristic
45#4532C2212yes—
46#4530C2212yes—
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2912D311yescharacteristic, complemented
51#1972—31no—
52#1951—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1773—31no—
57#1765—31no—
58C2×C3×#41C2311yescharacteristic
59C22×D12C2312yes—
60C3×#131—22no—
61C3×#131C2×C414yescomplemented
62C2×C3×M16—21no—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8—21no—
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4—42no—
69C3×#117—22no—
70C3×#117C2312yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73#949—24no—
74#949C2×C418yescomplemented
75C2×#215—22no—
76C2×#213—22no—
77C2×(C2×D3)⋊C4—42no—
78#926C2311yescharacteristic
79C2×C4×Dic3C2311yescharacteristic
80C2×#208C2311yescharacteristic
81C2×#208C2312yes—
82C22×C3⋊C8C2311yescharacteristic
83C2×D24—21no—
84C2×D24—21no—
85C2×D24—22no—
86C2×D24C2×C414yescomplemented
87#893—22no—
88#893—22no—
89D3×C22⋊C4—42no—
90C4×D12—22no—
91C4×D12—22no—
92C3×#107—22no—
93C3×#107D412yes—
94C3×#106—21no—
95C3×#106—21no—
96C3×#106—22no—
97C3×#106D411yescharacteristic
98C3×#106D411yescharacteristic
99#837—22no—
100#835D412yes—
101#832D412yes—
102#823C2×C418yescomplemented
103#819C2×C414yescomplemented
104#815—21no—
105#815—22no—
106C8×Dic3—21no—
107#809—22no—
108#809—22no—
109#809—22no—
110#809—22no—
111#808—22no—
112#807—22no—
113#803—22no—
114C4×C3⋊C8—22no—
115C2×AGL(1,ℤ/8)—32no—
116C22×D8—31no—
117C2×#131C2×D311yescharacteristic
118#520C2×D312yes—
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—22no—
130#428—32no—
131#428C2×D312yes—
132#427—21no—
133#427C2×D311yescharacteristic
134C2×#106—21no—
135C2×#106C2×D311yescharacteristic
136C2×#104—31no—
137#418—32no—
138#416—31no—
139C2×#102—32no—
140C2×#100—32no—
141#405—31no—
142#405—32no—
143C2×#97—31no—
144#390—32no—
145#343—32no—
146#341—31no—
147#339—31no—
148#339—32no—
149C23×D3—22no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41C22×C412yes—
153C3×#41#4112yes—
154C2×C3×Q8C2×D411yescharacteristic
155C2×C3×D4—21no—
156C2×C3×D4C2×D411yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12C22×C414yes—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C3×M16—21no—
171C3×M16—24no—
172C3×M16C22×C412yes—
173C2×C3×C8—21no—
174C2×C3×C8—24no—
175C2×C3×C8C22×C412yes—
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8#4112yes—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—42no—
186C3×C42—22no—
187C3×C42—42no—
188#217—21no—
189#215—28no—
190#213—28no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—84no—
194C4⋊Dic3C2×D412yes—
195C4×Dic3—21no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198#208—22no—
199#208—21no—
200#208—21no—
201#208C22×C414yes—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8C2×D412yes—
207D24—24no—
208D24—24no—
209C2×#41C22×D311yescharacteristic
210C22×D4—32no—
211AGL(1,ℤ/8)—38no—
212AGL(1,ℤ/8)—64no—
213C2×SD16—62no—
214C2×D8—34no—
215C2×D8—62no—
216C2×D8—62no—
217C2×D8—61no—
218C2×D8—61no—
219#131—22no—
220#131C4×D314yescomplemented
221C2×M16—21no—
222C2×M16—32no—
223C2×M16—31no—
224C2×M16C22×D311yescharacteristic
225C22×C8—21no—
226C22×C8—31no—
227C22×C8C22×D311yescharacteristic
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—42no—
231C4×D4—62no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—62no—
235#117—22no—
236#117—31no—
237#117C22×D312yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22×D312yes—
240C2×C22⋊C4—122no—
241C2×C22⋊C4—122no—
242C2×C42—31no—
243#108—34no—
244#107—22no—
245#107—62no—
246#107#6712yes—
247#106—21no—
248#106—21no—
249#106—22no—
250#106—62no—
251#106#6711yescharacteristic
252#106#6711yescharacteristic
253#104—62no—
254#103—32no—
255#102—32no—
256#102—62no—
257#102—62no—
258#102—62no—
259#102—62no—
260#100—38no—
261#97—62no—
262#97—62no—
263#97—61no—
264C4×C8—62no—
265C4×C8—61no—
266C23×C3—21no—
267C23×C3H(5,𝔽2)11yescharacteristic
268C22×D3—24no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—82no—
272C22×D3—82no—
273C3×Q8—21no—
274C3×Q8C4×D412yes—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—21no—
278C3×D4C4×D412yes—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—82no—
286C2×C3×C4—162no—
287C2×C3×C4C23×C411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296D12—24no—
297D12—24no—
298D12—42no—
299D12—42no—
300C4×D3—162no—
301C4×D3—162no—
302C3×C8—22no—
303C3×C8C4×D412yes—
304C3×C8C4×D412yes—
305C3⋊C8—24no—
306C3⋊C8—22no—
307C3⋊C8—41no—
308C24—62no—
309#41—21no—
310#41—21no—
311#41C2×C4×D312yes—
312#41#23912yes—
313C2×Q8D3×D411yescharacteristic
314C2×D4—21no—
315C2×D4—34no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4D3×D411yescharacteristic
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—61no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C23×D311yescharacteristic
329SD16—68no—
330D8—68no—
331D8—64no—
332D8—64no—
333M16—21no—
334M16—24no—
335M16—34no—
336M16—62no—
337M16—61no—
338M16—61no—
339M16C2×C4×D312yes—
340C2×C8—21no—
341C2×C8—24no—
342C2×C8—32no—
343C2×C8—61no—
344C2×C8—62no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8C2×C4×D312yes—
348C2×C8#23512yes—
349C2×C8C2×#6711yescharacteristic
350C2×C8C2×#6711yescharacteristic
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—32no—
354C4⋊C4—42no—
355C4⋊C4C2×#6712yes—
356C4⋊C4C2×#6712yes—
357C22⋊C4—22no—
358C22⋊C4—42no—
359C22⋊C4—61no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—244no—
363C42—22no—
364C42—42no—
365C42—61no—
366C42—61no—
367C42—61no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—42no—
371C22×C3#52011yescharacteristic
372C22×C3D4211yescharacteristic
373C22×C3#55311yescharacteristic
374C2×D3—44no—
375C2×D3—82no—
376C2×D3—82no—
377C2×D3—84no—
378C2×D3—162no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—82no—
383C3×C4C2×C4×D411yescharacteristic
384C3×C4C2×C4×D411yescharacteristic
385C3×C4#54711yescharacteristic
386C3×C4#54811yescharacteristic
387Dic3—42no—
388Dic3—81no—
389C23—21no—
390C23—64no—
391C23—122no—
392C23—122no—
393C23—242no—
394C23—242no—
395C23#100411yescharacteristic
396Q8—21no—
397Q8#89312yes—
398D4—22no—
399D4—22no—
400D4—21no—
401D4—64no—
402D4—64no—
403D4—122no—
404D4—122no—
405D4#89312yes—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—82no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—162no—
417C2×C4—482no—
418C2×C4—482no—
419C2×C4C22×C4×D311yescharacteristic
420C2×C4C2×#23511yescharacteristic
421C2×C4C2×D3×D411yescharacteristic
422C2×C4C2×#23911yescharacteristic
423C2×C4#100711yescharacteristic
424C2×C4C22×#6711yescharacteristic
425C8—22no—
426C8—64no—
427C8—62no—
428C8—121no—
429C8C4×#6712yes—
430C8C4×#6712yes—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—42no—
434C2×C3#236111yescharacteristic
435C2×C3#328812yes—
436D3—84no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—124no—
441C22—244no—
442C22—242no—
443C22—242no—
444C22—482no—
445C22#550411yescharacteristic
446C22#551411yescharacteristic
447C22D4×#6711yescharacteristic
448C4—22no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—122no—
453C4—241no—
454C4C2×#89311yescharacteristic
455C4#550711yescharacteristic
456C4C2×C4×#6711yescharacteristic
457C4#581711yescharacteristic
458C3#2304811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—42no—
462C2—244no—
463C2#8216311yescharacteristic
464C2#8401512yes—
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2912C2
socleC22×C3D42
prime radicalC2×#5142C2
cosocleC22×C3×C4C24
hypercenter#2912D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. C3×#2912C2semidirect
3. #82564C2semidirect
4. #79167C2semidirect
5. #72009C2semidirect
6. #71667C2semidirect
7. #70674C2semidirect
8. #70673C2semidirect
9. #70660C2semidirect
10. #70576C2semidirect
11. #70572C2semidirect
26. #5142C4semidirect
50. #2912D3semidirect
61. C3×#131C2×C4semidirect
74. #949C2×C4semidirect
86. C2×D24C2×C4semidirect
102. #823C2×C4semidirect
103. #819C2×C4semidirect
220. #131C4×D3semidirect
458. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C3×#29122
C2×#51422
#825642
#791672
#720092
#716672
#706742
#706732
#706602
#705762
#705722
#230483