Finite Group Databasebeta

← #10897653

Subgroups — #10897653

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2921C211yesmaximal, characteristic, complemented
2#78541C211yesmaximal, characteristic, complemented
3#71591C212yesmaximal, complemented
4#70531C211yesmaximal, characteristic, complemented
5#70530C212yesmaximal, complemented
6#70526C211yesmaximal, characteristic, complemented
7#69993C211yesmaximal, characteristic
8#69992C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#69874C212yesmaximal, complemented
11#69871C211yesmaximal, characteristic, complemented
12#29385—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14#5791C2212yes—
15#5748C2211yescharacteristic
16#5745C2212yes—
17C3×#433—22no—
18C3×#433—22no—
19C3×#423C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#407C2211yescharacteristic
22C3×#407C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24C3×C4×M16C2211yescharacteristic
25#5207C2212yes—
26#5139C2212yes—
27#5132C2212yes—
28#5125C2211yescharacteristic
29#5124C2212yes—
30C2×#805C2212yes—
31C2×#805C2212yes—
32#4717—22no—
33#4700—22no—
34C3⋊#338C2211yescharacteristic
35#4559C2212yes—
36#4557C2211yescharacteristic
37#4556C2212yes—
38#4523C2212yes—
39#4523C2212yes—
40#2956—31no—
41#2921D311yescharacteristic, complemented
42#2102—31no—
43#2091—31no—
44#2050—31no—
45#2010—31no—
46#1932—32no—
47#1897—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C23×C3×C4C2311yescharacteristic
52C2×#235C2312yes—
53C2×C3×M16C2311yescharacteristic
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2311yescharacteristic
57C3×#117—21no—
58C3×#117—21no—
59C3×#117—22no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64#953—22no—
65#930—22no—
66#929—22no—
67C4×#67—42no—
68#926C2312yes—
69#924—22no—
70C2×#208C2312yes—
71C2×#208C2312yes—
72#909—22no—
73#908—22no—
74#907—42no—
75D3×M16—42no—
76#904—22no—
77#901—24no—
78C3×C4⋊C8—22no—
79C3×C4⋊C8—22no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×#97—21no—
83C3×#97D412yescomplemented
84C3×C4×C8—21no—
85C3×C4×C8D412yescomplemented
86#830—42no—
87#825—42no—
88#824—22no—
89#823—22no—
90#822—42no—
91#819—22no—
92#805—22no—
93#805—22no—
94#805—24no—
95#575—31no—
96#568—32no—
97#525—32no—
98C2×#117C2×D311yescharacteristic
99#444—62no—
100#443—62no—
101#433—22no—
102#433—22no—
103#430—31no—
104#430—32no—
105#423C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107#421—32no—
108C2×#104—32no—
109C2×#104—32no—
110#413—32no—
111#407—32no—
112#407C2×D311yescharacteristic
113#407C2×D311yescharacteristic
114C4×M16C2×D311yescharacteristic
115C4×M16C2×D311yescharacteristic
116#344—32no—
117#344—32no—
118#339—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)—31no—
122C24×C3#4111yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4C2411yescharacteristic
131C2×#67—22no—
132C2×#67—42no—
133#235—22no—
134#235—22no—
135#235—22no—
136#235—22no—
137C2×D12#4112yes—
138C2×C4×D3—22no—
139C2×Dic6#4112yes—
140C3×M16—22no—
141C3×M16—22no—
142C3×M16—22no—
143C3×M16—22no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8C2×D412yes—
147C2×C3×C8C2×D412yes—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4112yes—
157#217—22no—
158#217—42no—
159(C2×D3)⋊C4—42no—
160C4⋊Dic3—22no—
161Dic3⋊C4—42no—
162C4×Dic3—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166#208—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169Dic12—22no—
170D24—22no—
171#204—24no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C2×#41—32no—
175C23×C4C22×D311yescharacteristic
176#137—62no—
177AGL(1,ℤ/8)—62no—
178#135—62no—
179#131—64no—
180#131—122no—
181C2×M16—32no—
182C2×M16—32no—
183C2×M16—122no—
184C2×M16C22×D311yescharacteristic
185C2×M16C22×D311yescharacteristic
186C2×M16C22×D311yescharacteristic
187C2×M16C22×D311yescharacteristic
188#123—62no—
189C22⋊Q8—62no—
190#121—62no—
191#120—62no—
192C4×D4—122no—
193#117—22no—
194#117—21no—
195#117—21no—
196#117—32no—
197C2×C4⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42C22×D311yescharacteristic
200C2×C42C22×D311yescharacteristic
201#108—62no—
202#108—122no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205#104—64no—
206#104—62no—
207#104—62no—
208#104—122no—
209#101—62no—
210#100—62no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—122no—
214#97—21no—
215#97#6712yescomplemented
216C4×C8—21no—
217C4×C8#6712yescomplemented
218C23×C3—21no—
219C23×C3—22no—
220C23×C3—81no—
221C23×C3C2×#4111yescharacteristic
222C22×D3—22no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4C22×D411yescharacteristic
234C2×C3×C4C2×#4111yescharacteristic
235C2×C3×C4C2×#4112yes—
236C2×C3×C4H(5,𝔽2)12yes—
237#67—42no—
238#67—42no—
239#67—84no—
240C2×Dic3—22no—
241C2×Dic3—42no—
242D12—22no—
243D12—22no—
244C4×D3—42no—
245C4×D3—42no—
246Dic6—22no—
247Dic6—22no—
248C3×C8—22no—
249C3×C8—22no—
250C3×C8—41no—
251C3×C8—41no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C24#23511yescharacteristic
255#41—62no—
256#41—62no—
257#41—62no—
258#41—62no—
259C2×Q8—32no—
260C2×D4—32no—
261C2×D4—62no—
262C2×D4—122no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—62no—
270C22×C4#23511yescharacteristic
271C22×C4C23×D311yescharacteristic
272Q16—62no—
273SD16—64no—
274D8—62no—
275M16—22no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—62no—
283M16—124no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—124no—
289C2×C8C2×#6712yes—
290C2×C8C2×#6712yes—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—62no—
294C4⋊C4—122no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—62no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C42—21no—
301C42—21no—
302C42—62no—
303C42#23512yes—
304C42C2×#6711yescharacteristic
305C42C2×#6711yescharacteristic
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3#52511yescharacteristic
312C22×C3#55112yes—
313C2×D3—42no—
314C2×D3—82no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—42no—
318C3×C4—41no—
319C3×C4#52511yescharacteristic
320C3×C4#53511yescharacteristic
321C3×C4#55312yes—
322Dic3—42no—
323Dic3—82no—
324C23—22no—
325C23—21no—
326C23—62no—
327C23—81no—
328C23C2×#23511yescharacteristic
329Q8—62no—
330Q8—62no—
331D4—62no—
332D4—62no—
333D4—122no—
334D4—122no—
335D4—244no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—62no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4C2×#23511yescharacteristic
351C2×C4D3×#4112yes—
352C2×C4#100912yes—
353C2×C4C22×#6711yescharacteristic
354C8—22no—
355C8—22no—
356C8—41no—
357C8—41no—
358C8—122no—
359C8—122no—
360C2×C3—22no—
361C2×C3—21no—
362C2×C3—81no—
363C2×C3#259811yescharacteristic
364D3—82no—
365C22—41no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—122no—
371C22—242no—
372C22#552712yes—
373C22#579111yescharacteristic
374C4—42no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—122no—
379C4—242no—
380C4#553012yes—
381C4#582811yescharacteristic
382C4#583011yescharacteristic
383C3#2938511yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—81no—
387C2—242no—
388C2#8228411yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2921C2
socleC2×C3#2598
prime radical#69993C2
hypercenter#2921D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. C3×#2921C2semidirect
2. #78541C2semidirect
3. #71591C2semidirect
4. #70531C2semidirect
5. #70530C2semidirect
6. #70526C2semidirect
8. #69992C2semidirect
9. #69987C2semidirect
10. #69874C2semidirect
11. #69871C2semidirect
41. #2921D3semidirect
83. C3×#97D4semidirect
85. C3×C4×C8D4semidirect
215. #97#67semidirect
217. C4×C8#67semidirect
383. C3#29385semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#785412
#715912
#705312
#705302
#705262
#699932
#699922
#699872
#698742
#698712
#293853