Finite Group Databasebeta

← #10948635

Subgroups — #10948635

Order 768.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2943C211yesmaximal, characteristic, complemented
2#83894C211yesmaximal, characteristic
3C2×#4717C211yesmaximal, characteristic, complemented
4#72081C211yesmaximal, characteristic, complemented
5#70918C214yesmaximal, complemented
6#70917C214yesmaximal, complemented
7#70656C211yesmaximal, characteristic, complemented
8#70650C211yesmaximal, characteristic, complemented
9C2×#4466C211yesmaximal, characteristic, complemented
10#33468—31nomaximal, Sylow
11C2×C3×#135C2211yescharacteristic
12C3×C4×#41—21no—
13C2×#904C2211yescharacteristic
14C2×#901—21no—
15C3×#442C2212yescomplemented
16C3×#441C2212yescomplemented
17C3×#440—22no—
18C3×#427C2211yescharacteristic
19C3×#416—21no—
20C2×C3×#97C2211yescharacteristic
21#5195C412yescomplemented
22#5194C414yescomplemented
23#5193C412yescomplemented
24#5179C2212yes—
25#5147C2212yes—
26C2×#819C2211yescharacteristic
27C2×#812C2211yescharacteristic
28C2×#812C2211yescharacteristic
29C2×#805—21no—
30#4717—22no—
31#4717C2214yes—
32#4565C2212yescomplemented
33#4563C2212yescomplemented
34#4488C2214yes—
35#4487C2214yescomplemented
36#4466C2214yescomplemented
37#3414—31no—
38C2×#444—31no—
39#2943D311yescharacteristic, complemented
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42#2155—31no—
43#2148—31no—
44#2145—31no—
45C2×#347—31no—
46C2×C3×#41—21no—
47C2×#235—21no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135C2×C412yes—
51C3×#135C2×C412yes—
52C2×C3×Q16C2311yescharacteristic
53C2×C3×SD16—21no—
54C2×C3×D8C2311yescharacteristic
55C22×C3×C8D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×C4×Q8—22no—
58C3×C4×D4—22no—
59C3×C4×D4—42no—
60C3×#117—21no—
61C3×#117—41no—
62C2×C3×C42—41no—
63C2×C3×C42D411yescharacteristic
64C2×#208—21no—
65C2×Dic12C2×C411yescharacteristic, complemented
66#904—21no—
67#904—21no—
68#904C2×C412yescomplemented
69#904C2312yes—
70C2×D24C2×C411yescharacteristic, complemented
71C2×#204—21no—
72#901—24no—
73#901—42no—
74C2×ΓL(1,𝔽25)—41no—
75C2×D3×C8—41no—
76C3×#107—21no—
77C3×#106C2312yes—
78C3×#103—22no—
79C3×#102—22no—
80C3×#97D412yes—
81C3×#97C2312yes—
82#829C2×C414yescomplemented
83#828C2×C414yescomplemented
84#827C2×C414yescomplemented
85#826C2×C414yescomplemented
86#819—21no—
87#819C2312yes—
88#812—21no—
89#812C2×C412yescomplemented
90#812C2312yes—
91#812C2312yes—
92C2×C3⋊C16—21no—
93C2×C3⋊C16C2×C412yescomplemented
94#805—24no—
95C2×#135—31no—
96C2×#135C2×D311yescharacteristic
97C2×#131—61no—
98C4×#41—21no—
99#510—32no—
100#510—32no—
101#509—32no—
102#509—32no—
103#508—34no—
104C2×M32—31no—
105C2×M32—31no—
106#444—34no—
107#444—62no—
108#442C2×D312yescomplemented
109#441C2×D312yescomplemented
110#440—22no—
111C2×#108—31no—
112#427C2×D311yescharacteristic
113C2×#104—61no—
114#416—21no—
115C2×#97C2×D311yescharacteristic
116#365—34no—
117#362—32no—
118#361—32no—
119#360—34no—
120#347—34no—
121C3×#41—22no—
122C3×#41—22no—
123C3×#41—41no—
124C3×#41—41no—
125C2×C3×Q8—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—42no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—41no—
133#235—22no—
134#235—22no—
135#235—41no—
136#235—41no—
137C2×D12—21no—
138C2×C4×D3—41no—
139C2×Dic6—21no—
140C3×Q16—21no—
141C3×Q16C22×C412yes—
142C3×SD16—22no—
143C3×SD16—22no—
144C3×D8—21no—
145C3×D8C22×C412yes—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8C22×C411yescharacteristic
149C2×C3×C8C22×C411yescharacteristic
150C2×C3×C8C2×D412yes—
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8#4111yescharacteristic
153C2×C3×C8#4111yescharacteristic
154C2×C3×C8C2411yescharacteristic
155C3×C4⋊C4—22no—
156C3×C4⋊C4—42no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—41no—
159C3×C42—21no—
160C3×C42—42no—
161C3×C42—41no—
162C3×C42C2×D412yes—
163#208—22no—
164#208—41no—
165C2×C3⋊C8—41no—
166Dic12—21no—
167Dic12C22×C412yes—
168D24—21no—
169D24C22×C412yes—
170#204—22no—
171#204—22no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C3⋊C16—22no—
175C3⋊C16C22×C414yes—
176C2×#41—21no—
177C2×#41—61no—
178#135—21no—
179#135—21no—
180#135—32no—
181#135—32no—
182#135—61no—
183#135—61no—
184#135C4×D312yes—
185#135C4×D312yes—
186C2×Q16—31no—
187C2×Q16C22×D311yescharacteristic
188C2×SD16—21no—
189C2×SD16—61no—
190C2×D8—31no—
191C2×D8C22×D311yescharacteristic
192#131—64no—
193#131—122no—
194C2×M16—61no—
195C2×M16—121no—
196C22×C8—121no—
197C22×C8#6711yescharacteristic
198C22×C8C22×D311yescharacteristic
199C4×Q8—22no—
200C4×D4—22no—
201C4×D4—42no—
202#117—21no—
203#117—41no—
204C2×C42—41no—
205C2×C42#6711yescharacteristic
206Q32—34no—
207SD32—34no—
208SD32—34no—
209D16—34no—
210M32—32no—
211M32—32no—
212M32—32no—
213M32—61no—
214C2×C16—32no—
215C2×C16—61no—
216#108—32no—
217#108—61no—
218#107—21no—
219#106C22×D312yes—
220#104—64no—
221#103—22no—
222#102—22no—
223#97#6712yes—
224#97C22×D312yes—
225C23×C3—41no—
226C23×C3AGL(1,ℤ/8)11yescharacteristic
227C22×D3—41no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×D4—22no—
231C3×D4—42no—
232C3×D4—42no—
233C3×D4—41no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—81no—
240C2×C3×C4—84no—
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4C22×D411yescharacteristic
247#67—42no—
248#67—42no—
249C2×Dic3—41no—
250D12—22no—
251D12—41no—
252C4×D3—42no—
253C4×D3—42no—
254Dic6—22no—
255Dic6—41no—
256C3×C8—22no—
257C3×C8C4×D411yescharacteristic
258C3×C8C4×D411yescharacteristic
259C3×C8C23×C411yescharacteristic
260C3×C8C2×#4111yescharacteristic
261C3⋊C8—42no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266#41—62no—
267#41—62no—
268#41—121no—
269#41—121no—
270C2×Q8—21no—
271C2×Q8—61no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—61no—
275C2×D4—121no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—121no—
280C22×C4C2×#6711yescharacteristic
281Q16—21no—
282Q16—32no—
283Q16—61no—
284Q16C2×C4×D312yes—
285SD16—22no—
286SD16—22no—
287SD16—62no—
288SD16—62no—
289D8—21no—
290D8—32no—
291D8—61no—
292D8C2×C4×D312yes—
293M16—62no—
294M16—124no—
295M16—121no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—124no—
299C2×C8—121no—
300C2×C8C2×C4×D311yescharacteristic
301C2×C8C2×C4×D311yescharacteristic
302C2×C8#23511yescharacteristic
303C2×C8#23511yescharacteristic
304C2×C8C2×#6712yes—
305C2×C8C2×#6711yescharacteristic
306C2×C8C23×D311yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—41no—
311C42—21no—
312C42—42no—
313C42—41no—
314C42C2×#6712yes—
315C16—34no—
316C16—62no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3#44211yescharacteristic
321C22×C3#44211yescharacteristic
322C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
323C2×D3—42no—
324C2×D3—82no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4—82no—
328C3×C4#44211yescharacteristic
329C3×C4#44211yescharacteristic
330C3×C4C2×C4×D411yescharacteristic
331C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
332Dic3—42no—
333C23—41no—
334C23—121no—
335C23#93211yescharacteristic
336Q8—22no—
337Q8—41no—
338Q8—62no—
339Q8—121no—
340D4—22no—
341D4—42no—
342D4—42no—
343D4—41no—
344D4—62no—
345D4—122no—
346D4—122no—
347D4—121no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—84no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4C4×#6711yescharacteristic
359C2×C4C4×#6711yescharacteristic
360C2×C4#93211yescharacteristic
361C2×C4#94911yescharacteristic
362C2×C4#94911yescharacteristic
363C2×C4C22×#6711yescharacteristic
364C8—22no—
365C8—122no—
366C8C4×#6711yescharacteristic
367C8C4×#6711yescharacteristic
368C8C22×C4×D311yescharacteristic
369C8C2×#23511yescharacteristic
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—82no—
374C2×C3C2×#44211yescharacteristic
375D3—82no—
376C22—41no—
377C22—42no—
378C22—82no—
379C22—122no—
380C22—242no—
381C22#501411yescharacteristic
382C22#501411yescharacteristic
383C22C2×#93211yescharacteristic
384C4—42no—
385C4—42no—
386C4—82no—
387C4—122no—
388C4#501411yescharacteristic
389C4#501411yescharacteristic
390C4C2×C4×#6711yescharacteristic
391C4C2×#94911yescharacteristic
392C3#3346811yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—242no—
398C2C2×#501411yescharacteristic
subgroupG/H
centerC4#5014
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2943C2
socleC2×C3C2×#442
prime radical#83894C2
cosocleC2×C3×C8C24
hypercenter#2943D3
hypocenterC3#33468
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ #2943

Complements

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

NH ≅ G/Ntype
1. C3×#2943C2semidirect
3. C2×#4717C2semidirect
4. #72081C2semidirect
5. #70918C2semidirect
6. #70917C2semidirect
7. #70656C2semidirect
8. #70650C2semidirect
9. C2×#4466C2semidirect
15. C3×#442C22semidirect
16. C3×#441C22semidirect
21. #5195C4semidirect
22. #5194C4semidirect
23. #5193C4semidirect
32. #4565C22semidirect
33. #4563C22semidirect
35. #4487C22semidirect
36. #4466C22semidirect
39. #2943D3semidirect
65. C2×Dic12C2×C4semidirect
68. #904C2×C4semidirect
70. C2×D24C2×C4semidirect
82. #829C2×C4semidirect
83. #828C2×C4semidirect
84. #827C2×C4semidirect
85. #826C2×C4semidirect
89. #812C2×C4semidirect
93. C2×C3⋊C16C2×C4semidirect
108. #442C2×D3semidirect
109. #441C2×D3semidirect
392. C3#33468semidirect

Maximal subgroups

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

subgroupindex
C3×#29432
#838942
C2×#47172
#720812
#709182
#709172
#706562
#706502
C2×#44662
#334683