Finite Group Databasebeta

← #10897656

Subgroups — #10897656

Order 768.

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

normal subgroups only

This group has 662 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#79039C211yesmaximal, characteristic, complemented
3C3×#2102C211yesmaximal, characteristic, complemented
4#70729C211yesmaximal, characteristic, complemented
5#70531C211yesmaximal, characteristic, complemented
6#69036C212yesmaximal, complemented
7#69012C212yesmaximal, complemented
8#68773C212yesmaximal, complemented
9#68757C212yesmaximal, complemented
10#68703C211yesmaximal, characteristic
11#68696C211yesmaximal, characteristic, complemented
12#29385—31nomaximal, Sylow
13#5748C2211yescharacteristic
14#5745C2212yes—
15#5524C2212yes—
16C2×#872C2211yescharacteristic
17C3×#430C2211yescharacteristic
18C3×#423C2211yescharacteristic
19C2×#825C2212yes—
20#5132C2212yes—
21#5126C2211yescharacteristic
22#5126C2211yescharacteristic
23#5126C2212yes—
24#5124C2212yes—
25Dic3×M16C2211yescharacteristic
26Dic3×M16C2211yescharacteristic
27#5005C2212yes—
28C2×#805C2212yes—
29#4924C2211yescharacteristic
30#4895—22no—
31#4894—22no—
32#4840—22no—
33#4840—22no—
34C3×#344C2212yes—
35C3×#339C2212yes—
36C3×H(3,ℤ/4)C2211yescharacteristic
37C3⋊#338C2211yescharacteristic
38#4533C2212yes—
39#4532C2212yes—
40#2956—31no—
41#2921—31no—
42#2102D311yescharacteristic, complemented
43#2091—31no—
44#2050—31no—
45#2010—31no—
46#1932—32no—
47#1897—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C2×#235C2312yes—
52C22×C4×D3C2311yescharacteristic
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58C2×(C2×D3)⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3C2311yescharacteristic
61C2×#208C2311yescharacteristic
62C2×#208C2312yes—
63C2×#208C2311yescharacteristic
64#909—22no—
65#908—22no—
66#907—24no—
67D3×M16—42no—
68#904—22no—
69#901—42no—
70#897—22no—
71#895—22no—
72#886—22no—
73#882—22no—
74C4×D12—42no—
75#872—22no—
76#872—21no—
77#872—21no—
78C3×#108—22no—
79C3×C4⋊C8—22no—
80#825—22no—
81#825—22no—
82#824—22no—
83#823—22no—
84#822—42no—
85(C2×D3)⋊C8—22no—
86(C2×D3)⋊C8—22no—
87(C2×D3)⋊C8—42no—
88#815—21no—
89#815D412yescomplemented
90C8×Dic3—21no—
91C8×Dic3D412yescomplemented
92#805—24no—
93#805—42no—
94C4⋊(C3⋊C8)—22no—
95#575—31no—
96#568—32no—
97#525—32no—
98C2×#117—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#433—62no—
103#430—32no—
104#430C2×D311yescharacteristic
105#423—31no—
106#423C2×D311yescharacteristic
107#421—32no—
108C2×#104—32no—
109C2×#104—32no—
110#413—32no—
111#407—31no—
112#407—31no—
113#407—32no—
114C4×M16—31no—
115C4×M16—31no—
116#344—32no—
117#344C2×D312yes—
118#339—32no—
119#339C2×D312yes—
120#338—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C23×D3#4111yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4C2411yescharacteristic
125C2×#67—22no—
126C22×Dic3—21no—
127C22×Dic3#4111yescharacteristic
128#235—22no—
129#235—22no—
130#235—22no—
131#235—22no—
132C2×D12—42no—
133C2×D12#4112yes—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—22no—
137C2×C4×D3—22no—
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—22no—
146C2×C3×C8C2×D412yes—
147C3×C4⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C42—21no—
150C3×C42—22no—
151C3×C42#4112yes—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—42no—
155(C2×D3)⋊C4—42no—
156C4⋊Dic3—42no—
157Dic3⋊C4—22no—
158Dic3⋊C4—22no—
159C4×Dic3—21no—
160C4×Dic3C2×D411yescharacteristic
161C4×Dic3C2×D411yescharacteristic
162#208—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8C2×D412yes—
169Dic12—22no—
170D24—22no—
171#204—24no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C2×#41—32no—
175C23×C4—31no—
176#137—62no—
177AGL(1,ℤ/8)—62no—
178#135—62no—
179#131—64no—
180#131—122no—
181C2×M16—31no—
182C2×M16—32no—
183C2×M16—31no—
184C2×M16—122no—
185C2×M16C22×D312yes—
186C2×M16C22×D311yescharacteristic
187C2×M16C22×D311yescharacteristic
188#123—62no—
189C22⋊Q8—62no—
190#121—62no—
191#120—62no—
192C4×D4—122no—
193#117—61no—
194#117—61no—
195#117—62no—
196#117C22×D312yes—
197C2×C4⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—31no—
200C2×C42C22×D311yescharacteristic
201#108—22no—
202#108—122no—
203C4⋊C8—22no—
204C4⋊C8—62no—
205#104—62no—
206#104—62no—
207#104—64no—
208#104—122no—
209#101—62no—
210#100—62no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—32no—
215#97—61no—
216C4×C8—32no—
217C4×C8—61no—
218C23×C3C2×#4111yescharacteristic
219C22×D3—21no—
220C22×D3—22no—
221C22×D3—22no—
222C22×D3—81no—
223C2×C3×C4—22no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4C22×D411yescharacteristic
227C2×C3×C4C2×#4111yescharacteristic
228C2×C3×C4C2×#4112yes—
229C2×C3×C4H(5,𝔽2)12yes—
230#67—42no—
231#67—42no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—81no—
238D12—22no—
239D12—22no—
240D12—84no—
241C4×D3—41no—
242C4×D3—42no—
243C4×D3—41no—
244C4×D3—42no—
245C4×D3—42no—
246Dic6—22no—
247Dic6—22no—
248C3×C8—22no—
249C3×C8—41no—
250C3×C8—42no—
251C3⋊C8—22no—
252C3⋊C8—41no—
253C3⋊C8—42no—
254C24—31no—
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—31no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
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—22no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—62no—
289C2×C8—124no—
290C2×C8D3×D412yes—
291C4⋊C4—22no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—122no—
295C22⋊C4—22no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C42—21no—
301C42—22no—
302C42—31no—
303C42—31no—
304C42—61no—
305C42#23912yes—
306C22×C3—41no—
307C22×C3#52511yescharacteristic
308C22×C3#55112yes—
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—41no—
313C2×D3—42no—
314C2×D3—82no—
315C3×C4—42no—
316C3×C4—82no—
317C3×C4#52511yescharacteristic
318C3×C4#53511yescharacteristic
319C3×C4#55312yes—
320Dic3—41no—
321Dic3—41no—
322Dic3—42no—
323Dic3—41no—
324C23—62no—
325C23—61no—
326C23—62no—
327C23—241no—
328C23D3×#4111yescharacteristic
329Q8—62no—
330Q8—62no—
331D4—62no—
332D4—62no—
333D4—122no—
334D4—122no—
335D4—244no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—81no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—241no—
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×#23911yescharacteristic
352C2×C4D3×#4112yes—
353C2×C4#100912yes—
354C8—22no—
355C8—41no—
356C8—42no—
357C8—62no—
358C8—121no—
359C8—122no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3#259811yescharacteristic
363D3—81no—
364D3—82no—
365C22—41no—
366C22—122no—
367C22—122no—
368C22—122no—
369C22—121no—
370C22—122no—
371C22—242no—
372C22#562711yescharacteristic
373C22#570512yes—
374C4—42no—
375C4—82no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4—122no—
380C4#565611yescharacteristic
381C4#567411yescharacteristic
382C4#569512yes—
383C3#2938511yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—241no—
387C2—242no—
388C2#7820311yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2102C2
socleC2×C3#2598
prime radical#68703C2
hypercenter#2102D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2102

Complements

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

NH ≅ G/Ntype
1. #79258C2semidirect
2. #79039C2semidirect
3. C3×#2102C2semidirect
4. #70729C2semidirect
5. #70531C2semidirect
6. #69036C2semidirect
7. #69012C2semidirect
8. #68773C2semidirect
9. #68757C2semidirect
11. #68696C2semidirect
42. #2102D3semidirect
89. #815D4semidirect
91. C8×Dic3D4semidirect
383. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#792582
#790392
C3×#21022
#707292
#705312
#690362
#690122
#687732
#687572
#687032
#686962
#293853