Finite Group Databasebeta

← #10795556

Subgroups — #10795556

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78550C211yesmaximal, characteristic
2C3×#1747C211yesmaximal, characteristic, complemented
3#69898C212yesmaximal
4#69895C211yesmaximal, characteristic, complemented
5#69890C212yesmaximal, complemented
6#68724C211yesmaximal, characteristic
7#68716C211yesmaximal, characteristic, complemented
8#68711C212yesmaximal
9#68710C211yesmaximal, characteristic
10#68699C212yesmaximal, complemented
11#68696C211yesmaximal, characteristic, complemented
12#20788—31nomaximal, Sylow
13C2×#907C2211yescharacteristic
14C2×#901C2211yescharacteristic
15#5522C2211yescharacteristic
16C3×#405C2212yes—
17C3×#405C2211yescharacteristic
18#5132C2212yes—
19C2×#822C2212yes—
20Dic3×M16C2211yescharacteristic
21#5101C2212yes—
22C2×#815C2211yescharacteristic
23#5006C2212yes—
24#4998C2212yes—
25#4997C2211yescharacteristic
26#4708C412yescomplemented
27D3×#97C412yescomplemented
28#4689C412yescomplemented
29C4×ΓL(1,𝔽25)C412yescomplemented
30C3×#338C2211yescharacteristic
31C3×H(3,ℤ/4)C2211yescharacteristic
32C3×H(3,ℤ/4)C2212yes—
33C3⋊#338C2211yescharacteristic
34#4559C2212yes—
35#4557C2211yescharacteristic
36#4555C2212yes—
37#4533C2212yes—
38#4532C2212yes—
39#4531C2212yes—
40#2875—31no—
41#1841—31no—
42#1841—31no—
43#1833—32no—
44#1827—32no—
45#1813—31no—
46#1812—31no—
47#1773—32no—
48#1757—32no—
49#1747—31no—
50#1747D311yescharacteristic, complemented
51C2×#235C2311yescharacteristic
52C2×C3×M16C2312yes—
53C2×C3×M16C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×#117C2312yes—
56C3×#117C2311yescharacteristic
57#926C2312yes—
58C2×C4×Dic3C2311yescharacteristic
59C2×#208C2312yes—
60C2×#208C2311yescharacteristic
61C22×C3⋊C8C2311yescharacteristic
62#907—22no—
63#907—22no—
64#901—22no—
65#901—22no—
66C2×ΓL(1,𝔽25)C2×C412yes—
67C2×ΓL(1,𝔽25)C2×C412yes—
68C2×D3×C8C2×C412yes—
69C2×D3×C8C2×C412yes—
70#893—21no—
71#887—21no—
72#67⋊C4—22no—
73#872C2×C412yes—
74C42×D3C2×C412yes—
75C3×#97—22no—
76C3×#97C2×C412yes—
77C3×C4×C8—22no—
78C3×C4×C8C2×C412yes—
79#824—22no—
80#823—22no—
81#822—22no—
82#822—22no—
83#815—22no—
84#815C2×C411yescharacteristic
85#815C2×C412yes—
86#815C2×C411yescharacteristic
87#815C2×C411yescharacteristic
88#815C2×C411yescharacteristic
89C8×Dic3—22no—
90C8×Dic3C2×C412yes—
91#808—22no—
92#807—22no—
93#806—24no—
94#803C2×C412yes—
95C4×C3⋊C8C2×C412yes—
96C2×#131—31no—
97C2×#131—31no—
98C4×#41—31no—
99C2×#108—32no—
100#427—32no—
101#413—32no—
102#410—32no—
103#405—31no—
104#405—32no—
105#405—32no—
106#405—32no—
107#405C2×D312yes—
108#405C2×D311yescharacteristic
109C4×M16—31no—
110C4×M16—32no—
111C2×#97—31no—
112C2×#97—32no—
113#344—32no—
114#344—32no—
115#339—32no—
116#339—32no—
117#338—31no—
118#338C2×D311yescharacteristic
119H(3,ℤ/4)—31no—
120H(3,ℤ/4)—32no—
121H(3,ℤ/4)C2×D311yescharacteristic
122H(3,ℤ/4)C2×D312yes—
123C22×C3×C4C2411yescharacteristic
124C2×#67#4112yes—
125C22×Dic3—21no—
126#235—22no—
127#235—22no—
128C2×D12#4111yescharacteristic
129C2×C4×D3—22no—
130C2×C4×D3C22×C412yes—
131C2×Dic6#4111yescharacteristic
132C3×M16—22no—
133C3×M16—22no—
134C3×M16—22no—
135C2×C3×C8—22no—
136C2×C3×C8C22×C412yes—
137C2×C3×C8C22×C411yescharacteristic
138C2×C3×C8C22×C411yescharacteristic
139C2×C3×C8#4112yes—
140C2×C3×C8#4112yes—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—22no—
145C3×C42—22no—
146C3×C42C22×C412yes—
147#217—22no—
148(C2×D3)⋊C4—22no—
149C4⋊Dic3—22no—
150Dic3⋊C4—22no—
151C4×Dic3—21no—
152C4×Dic3—22no—
153C4×Dic3—22no—
154C4×Dic3C22×C411yescharacteristic
155C4×Dic3C22×C411yescharacteristic
156#208—22no—
157#208—22no—
158#208—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8C22×C412yes—
161C2×C3⋊C8C22×C411yescharacteristic
162C2×C3⋊C8C22×C411yescharacteristic
163C2×C3⋊C8#4112yes—
164C2×C3⋊C8#4112yes—
165ΓL(1,𝔽25)—22no—
166ΓL(1,𝔽25)—22no—
167ΓL(1,𝔽25)—24no—
168D3×C8—24no—
169D3×C8—22no—
170D3×C8—22no—
171C2×#41—31no—
172#131—62no—
173#131—62no—
174#131—62no—
175#131—62no—
176C2×M16—31no—
177C2×M16—32no—
178C2×M16—32no—
179C2×M16—32no—
180C2×M16C22×D312yes—
181C2×M16C22×D311yescharacteristic
182C22×C8—31no—
183C22×C8—32no—
184C22×C8—32no—
185C22×C8C22×D311yescharacteristic
186C4×Q8—61no—
187C4×D4—62no—
188C4×D4—61no—
189#117—32no—
190#117—32no—
191#117C22×D312yes—
192#117C22×D311yescharacteristic
193C2×C42—31no—
194C2×C42—32no—
195#108—62no—
196#108—62no—
197#107—62no—
198#106—62no—
199#101—62no—
200#100—62no—
201#99—64no—
202#97—22no—
203#97—31no—
204#97—31no—
205#97—32no—
206#97—31no—
207#97—31no—
208#97—32no—
209#97—62no—
210#97C4×D312yes—
211C4×C8—22no—
212C4×C8—32no—
213C4×C8—32no—
214C4×C8—62no—
215C4×C8C4×D312yes—
216C23×C3C2×#4111yescharacteristic
217C22×D3#11712yes—
218C2×C3×C4—22no—
219C2×C3×C4—42no—
220C2×C3×C4C23×C411yescharacteristic
221C2×C3×C4C2×#4112yes—
222C2×C3×C4C2×#4112yes—
223C2×C3×C4C2×#4111yescharacteristic
224#67—24no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—42no—
228C2×Dic3—81no—
229C2×Dic3#11712yes—
230D12—22no—
231C4×D3—22no—
232C4×D3—22no—
233C4×D3—82no—
234Dic6—22no—
235C3×C8—22no—
236C3×C8—42no—
237C3×C8#11712yes—
238C3×C8#11711yescharacteristic
239C3×C8#11711yescharacteristic
240C3⋊C8—22no—
241C3⋊C8—42no—
242C3⋊C8#11711yescharacteristic
243C3⋊C8#11711yescharacteristic
244C3⋊C8#11712yes—
245#41—62no—
246#41—62no—
247C2×Q8—31no—
248C2×D4—32no—
249C2×D4—31no—
250C22×C4—32no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4C23×D311yescharacteristic
254M16—22no—
255M16—22no—
256M16—22no—
257M16—64no—
258M16—62no—
259M16—62no—
260M16—62no—
261M16—62no—
262M16—62no—
263C2×C8—22no—
264C2×C8—31no—
265C2×C8—31no—
266C2×C8—32no—
267C2×C8—32no—
268C2×C8—32no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—64no—
272C2×C8—62no—
273C2×C8C2×C4×D312yes—
274C2×C8C2×C4×D311yescharacteristic
275C2×C8C2×C4×D311yescharacteristic
276C2×C8#23512yes—
277C2×C8#23512yes—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C22⋊C4—21no—
283C22⋊C4—22no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C42—22no—
287C42—31no—
288C42—31no—
289C42—62no—
290C42—61no—
291C42—62no—
292C42C2×C4×D312yes—
293C22×C3—41no—
294C22×C3C2×#11711yescharacteristic
295C22×C3#52912yes—
296C2×D3—22no—
297C2×D3—42no—
298C3×C4—42no—
299C3×C4—82no—
300C3×C4C2×#11711yescharacteristic
301C3×C4C2×#11711yescharacteristic
302C3×C4#52912yes—
303Dic3—22no—
304Dic3—41no—
305Dic3—41no—
306Dic3—82no—
307C23—32no—
308C23D3×#4111yescharacteristic
309Q8—62no—
310D4—64no—
311D4—62no—
312C2×C4—22no—
313C2×C4—32no—
314C2×C4—42no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—122no—
320C2×C4—242no—
321C2×C4—241no—
322C2×C4C22×C4×D311yescharacteristic
323C2×C4C2×#23512yes—
324C2×C4D3×#4112yes—
325C2×C4D3×#4111yescharacteristic
326C8—22no—
327C8—31no—
328C8—31no—
329C8—32no—
330C8—42no—
331C8—62no—
332C8—122no—
333C8#87212yes—
334C8#87211yescharacteristic
335C8#87211yescharacteristic
336C2×C3—21no—
337C2×C3—22no—
338C2×C3#228311yescharacteristic
339D3—42no—
340C22—41no—
341C22—62no—
342C22—122no—
343C22D3×#11711yescharacteristic
344C22#552812yes—
345C4—42no—
346C4—62no—
347C4—82no—
348C4—121no—
349C4—121no—
350C4—242no—
351C4C2×#87211yescharacteristic
352C4D3×#11711yescharacteristic
353C4#552812yes—
354C3#2078811yescharacteristic, complemented, Sylow
355C2—21no—
356C2—22no—
357C2—122no—
358C2#7341311yescharacteristic
subgroupG/H
centerC4C2×#872
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1747C2
socleC2×C3#2283
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1747D3
hypocenterC3#20788
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
2. C3×#1747C2semidirect
4. #69895C2semidirect
5. #69890C2semidirect
7. #68716C2semidirect
10. #68699C2semidirect
11. #68696C2semidirect
26. #4708C4semidirect
27. D3×#97C4semidirect
28. #4689C4semidirect
29. C4×ΓL(1,𝔽25)C4semidirect
50. #1747D3semidirect
354. C3#20788semidirect

Maximal subgroups

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

subgroupindex
#785502
C3×#17472
#698982
#698952
#698902
#687242
#687162
#687112
#687102
#686992
#686962
#207883