Finite Group Databasebeta

← #10897643

Subgroups — #10897643

Order 768.

708 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79258C211yesmaximal, characteristic, complemented
2#78619C211yesmaximal, characteristic, complemented
3C3×#2091C211yesmaximal, characteristic, complemented
4#70735C212yesmaximal, complemented
5#70728C211yesmaximal, characteristic, complemented
6#70727C212yesmaximal, complemented
7#69993C211yesmaximal, characteristic
8#69992C212yesmaximal, complemented
9#69965C212yesmaximal, complemented
10#68719C211yesmaximal, characteristic, complemented
11#68716C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13#5791C2212yes—
14C22×C4×Dic3C2211yescharacteristic
15#5748C2211yescharacteristic
16#5745C2212yes—
17C3×#430C2211yescharacteristic
18C3×#430C2212yes—
19C3×#407C2211yescharacteristic
20C3×#407C2212yes—
21#5207C2211yescharacteristic
22#5139C2212yes—
23C2×#825C2212yes—
24C2×#825C2212yes—
25#5132C2212yes—
26#5118C2211yescharacteristic
27#5118C2211yescharacteristic
28Dic3×M16C2211yescharacteristic
29Dic3×M16C2211yescharacteristic
30#4895—22no—
31#4894—22no—
32#4719—22no—
33#4718—22no—
34C3×#338C2211yescharacteristic
35#4559C2212yes—
36#4557C2211yescharacteristic
37#4556C2212yes—
38#4555C2212yes—
39#4555C2212yes—
40#2956—31no—
41#2920—31no—
42#2102—31no—
43#2091D311yescharacteristic, complemented
44#2050—31no—
45#2010—31no—
46#1932—32no—
47#1896—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3—21no—
53C2×#235C2312yes—
54C2×C3×M16C2312yes—
55C2×C3×M16C2312yes—
56C2×C3×M16C2311yescharacteristic
57C2×C3×M16C2311yescharacteristic
58#953—22no—
59#930—22no—
60#929—22no—
61C4×#67—42no—
62#926C2312yes—
63#924—22no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3C2311yescharacteristic
70C2×C4×Dic3C2311yescharacteristic
71C2×#208C2311yescharacteristic
72C2×#208C2311yescharacteristic
73#909—22no—
74#908—22no—
75#907—24no—
76D3×M16—42no—
77#904—22no—
78#901—42no—
79C3×#108—22no—
80C3×#108—42no—
81C3×C22⋊C8—22no—
82C3×C22⋊C8—42no—
83#830—22no—
84#825—22no—
85#825—22no—
86#825—22no—
87#825—22no—
88#825—42no—
89#824—22no—
90#823—22no—
91#815—21no—
92#815D412yescomplemented
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95C8×Dic3—21no—
96C8×Dic3D412yescomplemented
97#575—31no—
98#568—32no—
99#525—32no—
100C22×C42—31no—
101#444—62no—
102#443—62no—
103#432—62no—
104#431—62no—
105#430C2×D311yescharacteristic
106#430C2×D312yes—
107#423—31no—
108#423—31no—
109#421—32no—
110C2×#104—32no—
111C2×#104—32no—
112#413—32no—
113#407—31no—
114#407C2×D311yescharacteristic
115#407C2×D312yes—
116C4×M16—31no—
117C4×M16—31no—
118#344—32no—
119#344—32no—
120#339—32no—
121#339—32no—
122#338C2×D311yescharacteristic
123H(3,ℤ/4)—31no—
124C24×C3#4111yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4C2411yescharacteristic
131C2×#67—22no—
132C2×#67—42no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—44no—
140#235—22no—
141#235—22no—
142#235—22no—
143#235—22no—
144C2×D12#4112yes—
145C2×C4×D3—22no—
146C2×Dic6#4112yes—
147C3×M16—22no—
148C3×M16—22no—
149C3×M16—22no—
150C3×M16—22no—
151C3×M16—22no—
152C3×M16—22no—
153C2×C3×C8—21no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8C2×D412yes—
157#217—22no—
158#217—42no—
159(C2×D3)⋊C4—42no—
160C4⋊Dic3—22no—
161Dic3⋊C4—42no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—22no—
169C4×Dic3C2×D411yescharacteristic
170C4×Dic3C2×D411yescharacteristic
171C4×Dic3#4111yescharacteristic
172C4×Dic3#4111yescharacteristic
173#208—22no—
174#208—22no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8C2×D412yes—
177Dic12—22no—
178D24—22no—
179#204—24no—
180ΓL(1,𝔽25)—44no—
181D3×C8—44no—
182C2×#41—32no—
183C23×C4—61no—
184C23×C4C22×D311yescharacteristic
185#137—62no—
186AGL(1,ℤ/8)—62no—
187#135—62no—
188#131—64no—
189#131—122no—
190C2×M16—31no—
191C2×M16—31no—
192C2×M16—122no—
193C2×M16C22×D312yes—
194C2×M16C22×D312yes—
195C2×M16C22×D311yescharacteristic
196C2×M16C22×D311yescharacteristic
197#123—62no—
198C22⋊Q8—62no—
199#121—62no—
200#120—62no—
201C4×D4—122no—
202#117—32no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210#108—22no—
211#108—42no—
212C4⋊C8—62no—
213C4⋊C8—62no—
214#104—62no—
215#104—62no—
216#104—62no—
217#104—62no—
218#104—122no—
219#101—62no—
220#100—62no—
221C22⋊C8—22no—
222C22⋊C8—42no—
223C22⋊C8—62no—
224#97—32no—
225#97—61no—
226C4×C8—32no—
227C4×C8—61no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3C2×#4111yescharacteristic
232C22×D3—22no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4C22×D411yescharacteristic
239C2×C3×C4C2×#4112yes—
240C2×C3×C4C2×#4111yescharacteristic
241C2×C3×C4H(5,𝔽2)12yes—
242#67—42no—
243#67—42no—
244#67—84no—
245C2×Dic3—24no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—22no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257D12—22no—
258D12—22no—
259C4×D3—42no—
260C4×D3—42no—
261Dic6—22no—
262Dic6—22no—
263C3×C8—22no—
264C3×C8—42no—
265C3×C8—41no—
266C3×C8—42no—
267C3⋊C8—22no—
268C3⋊C8—41no—
269C24#23911yescharacteristic
270#41—62no—
271#41—62no—
272#41—62no—
273#41—62no—
274C2×Q8—32no—
275C2×D4—32no—
276C2×D4—62no—
277C2×D4—122no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—124no—
290C22×C4#23911yescharacteristic
291C22×C4C23×D311yescharacteristic
292Q16—62no—
293SD16—64no—
294D8—62no—
295M16—22no—
296M16—22no—
297M16—22no—
298M16—22no—
299M16—22no—
300M16—22no—
301M16—62no—
302M16—62no—
303M16—124no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—32no—
308C2×C8—61no—
309C2×C8—124no—
310C2×C8D3×D412yes—
311C4⋊C4—62no—
312C4⋊C4—122no—
313C22⋊C4—62no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C42—31no—
317C42—31no—
318C42—31no—
319C42—31no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—61no—
325C42—61no—
326C42—62no—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3#52511yescharacteristic
334C22×C3#55112yes—
335C2×D3—42no—
336C2×D3—82no—
337C3×C4—41no—
338C3×C4#52511yescharacteristic
339C3×C4#53511yescharacteristic
340C3×C4#55312yes—
341Dic3—22no—
342Dic3—22no—
343Dic3—41no—
344Dic3—41no—
345Dic3—42no—
346Dic3—82no—
347C23—22no—
348C23—21no—
349C23—42no—
350C23—62no—
351C23C2×#23911yescharacteristic
352Q8—62no—
353Q8—62no—
354D4—62no—
355D4—62no—
356D4—122no—
357D4—122no—
358D4—244no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—64no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4C2×D3×D411yescharacteristic
379C2×C4#100412yes—
380C2×C4D3×#4112yes—
381C2×C4D3×#4111yescharacteristic
382C8—22no—
383C8—42no—
384C8—42no—
385C8—41no—
386C8—62no—
387C8—121no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—42no—
391C2×C3#259811yescharacteristic
392D3—82no—
393C22—22no—
394C22—24no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—122no—
400C22—242no—
401C22#562711yescharacteristic
402C22#563712yes—
403C4—41no—
404C4—62no—
405C4—62no—
406C4—121no—
407C4—121no—
408C4—122no—
409C4—242no—
410C4#565611yescharacteristic
411C4#566112yes—
412C4#571511yescharacteristic
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—42no—
417C2—242no—
418C2#7804411yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2091C2
socleC2×C3#2598
prime radical#69993C2
hypercenter#2091D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2091

Complements

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

NH ≅ G/Ntype
1. #79258C2semidirect
2. #78619C2semidirect
3. C3×#2091C2semidirect
4. #70735C2semidirect
5. #70728C2semidirect
6. #70727C2semidirect
8. #69992C2semidirect
9. #69965C2semidirect
10. #68719C2semidirect
11. #68716C2semidirect
43. #2091D3semidirect
92. #815D4semidirect
96. C8×Dic3D4semidirect
413. C3#29384semidirect

Maximal subgroups

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

subgroupindex
#792582
#786192
C3×#20912
#707352
#707282
#707272
#699932
#699922
#699652
#687192
#687162
#293843