Finite Group Databasebeta

← #10691701

Subgroups — #10691701

Order 768.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83216C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#71790C211yesmaximal, characteristic, complemented
5#70432C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#9269—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5110—21no—
15#5088—21no—
16#5084—22no—
17#5080—21no—
18#4968C414yes—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4838—21no—
23#4836—22no—
24#4834—21no—
25C3×#332—21no—
26C3×#332—21no—
27#4530—21no—
28#4530—21no—
29#4486—21no—
30#4485—21no—
31#3227—31no—
32#2935—31no—
33#1997—31no—
34#1547—31no—
35C2×#332D311yescharacteristic, complemented
36#1301—31no—
37#1283—31no—
38C22×C3×Q8—21no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C42C2311yescharacteristic
49#930—21no—
50#930C2×C412yes—
51C2×#208—21no—
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54#901—42no—
55#901—42no—
56C2×ΓL(1,𝔽25)—22no—
57C2×D3×C8—22no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12D412yes—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D411yescharacteristic
67C3×C4⋊C8D411yescharacteristic
68#830C2×C412yes—
69(C2×D3)⋊C8—22no—
70(C2×D3)⋊C8—22no—
71#809—24no—
72#808—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C2×#131—61no—
80#532—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#498—61no—
83#486—61no—
84#481—34no—
85#476—62no—
86#436—61no—
87#435—62no—
88C8×D4—61no—
89#431—31no—
90#423—31no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×C4⋊C8C2×D311yescharacteristic
93#408—61no—
94C4×M16—31no—
95#343—61no—
96#343—61no—
97#332—21no—
98#332—21no—
99#332—61no—
100#331—61no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—81no—
104C2×C3×Q8—81no—
105C22×C3×C4—22no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—22no—
110#235—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12C22⋊C412yes—
117C2×C4×D3—22no—
118C2×Dic6—21no—
119C2×C3×C8—21no—
120C2×C3×C8—21no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8—81no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3C22⋊C412yes—
137#208—21no—
138#208—21no—
139#208—21no—
140#208—41no—
141#208—41no—
142#208SD1612yes—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—41no—
147ΓL(1,𝔽25)—44no—
148D3×C8—44no—
149C2×#41—61no—
150C22×Q8—21no—
151#131—122no—
152#131—122no—
153C2×M16—31no—
154C2×M16—31no—
155C2×M16—62no—
156C2×M16—61no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—62no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165#127—31no—
166#124—61no—
167#121—32no—
168#121—61no—
169C4×D4—32no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C4×D312yes—
173C2×C42C22×D311yescharacteristic
174#106—64no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8#6711yescharacteristic
182C4⋊C8#6711yescharacteristic
183#102—64no—
184C22⋊C8—32no—
185C22⋊C8—62no—
186C22⋊C8—62no—
187#97—61no—
188C4×C8—61no—
189C4×C8—61no—
190C23×C3C4×D411yescharacteristic
191C22×D3—22no—
192C3×Q8—82no—
193C3×Q8—81no—
194C3×Q8—81no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—82no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4C2×SD1611yescharacteristic
207C2×C3×C4#13511yescharacteristic
208C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C3×C4#13711yescharacteristic
210#67—42no—
211#67—42no—
212C2×Dic3—22no—
213D12—42no—
214D12—41no—
215D12—41no—
216D12—41no—
217D12—41no—
218C4×D3—42no—
219C4×D3—42no—
220Dic6—41no—
221Dic6—41no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—81no—
225C3⋊C8—22no—
226C3⋊C8—42no—
227C3⋊C8—41no—
228C3⋊C8—81no—
229#41—121no—
230#41—121no—
231#41—121no—
232#41—121no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—61no—
236C2×Q8—81no—
237C2×Q8—81no—
238C2×D4—32no—
239C2×D4—62no—
240C2×D4—61no—
241C2×D4—61no—
242C22×C4—22no—
243C22×C4—62no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247M16—32no—
248M16—61no—
249M16—61no—
250M16—61no—
251M16—124no—
252M16—121no—
253M16—121no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—61no—
260C2×C8—62no—
261C2×C8—61no—
262C2×C8—81no—
263C2×C8—124no—
264C2×C8—121no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—32no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42C2×#6711yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3#38611yescharacteristic
280C22×C3C4×SD1611yescharacteristic
281C22×C3#44011yescharacteristic
282C2×D3—42no—
283C2×D3—42no—
284C2×D3—42no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—82no—
289C3×C4#45011yescharacteristic
290C3×C4#45211yescharacteristic
291C3×C4#46111yescharacteristic
292C3×C4#46311yescharacteristic
293C3×C4#47111yescharacteristic
294C3×C4#47211yescharacteristic
295Dic3—42no—
296C23—62no—
297C23#67⋊C411yescharacteristic
298Q8—82no—
299Q8—81no—
300Q8—81no—
301Q8—121no—
302Q8—121no—
303D4—122no—
304D4—122no—
305D4—122no—
306D4—121no—
307D4—121no—
308D4—121no—
309D4—121no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—62no—
315C2×C4—82no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4D3×C22⋊C411yescharacteristic
319C2×C4#88311yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4D3×SD1611yescharacteristic
322C2×C4#91411yescharacteristic
323C2×C4#91811yescharacteristic
324C2×C4#91911yescharacteristic
325C2×C4C4×#6711yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#93811yescharacteristic
328C8—41no—
329C8—41no—
330C8—62no—
331C8—81no—
332C8—122no—
333C8—121no—
334C8—241no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3#187711yescharacteristic
338C2×C3#190711yescharacteristic
339C2×C3#191511yescharacteristic
340D3—82no—
341C22—21no—
342C22—21no—
343C22—122no—
344C22—122no—
345C22—122no—
346C22#478811yescharacteristic
347C22#481711yescharacteristic
348C22#495611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—122no—
354C4#481811yescharacteristic
355C4#481911yescharacteristic
356C4#516011yescharacteristic
357C4#516911yescharacteristic
358C4#517711yescharacteristic
359C4#518711yescharacteristic
360C3#926911yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—242no—
364C2#7012411yescharacteristic
365C2#7021811yescharacteristic
366C2#7027611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9269
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. C2×C3×#332C2semidirect
4. #71790C2semidirect
6. #67707C2semidirect
7. #67706C2semidirect
35. C2×#332D3semidirect
360. C3#9269semidirect

Maximal subgroups

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

subgroupindex
#832162
#790042
C2×C3×#3322
#717902
#704322
#677072
#677062
#92693