Finite Group Databasebeta

← #10712106

Subgroups — #10712106

Order 768.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83792C211yesmaximal, characteristic
2C2×#4901C211yesmaximal, characteristic, complemented
3C2×C3×#350C211yesmaximal, characteristic, complemented
4#72007C211yesmaximal, characteristic, complemented
5#71438C211yesmaximal, characteristic, complemented
6#67813C211yesmaximal, characteristic, complemented
7#67805C211yesmaximal, characteristic, complemented
8#12987—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5822C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14#5236C411yescharacteristic
15#5145—22no—
16#5144—22no—
17#5139C411yescharacteristic
18#5124C411yescharacteristic
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C2×#812C2211yescharacteristic
22#4901—22no—
23#4901—22no—
24C2×#798—21no—
25C2×D3×C16—21no—
26C3×#350—21no—
27C3×#350—21no—
28#4553—22no—
29#4515—22no—
30C2×#504—31no—
31#2955—31no—
32#2117—31no—
33C2×#350D311yescharacteristic, complemented
34#2115—31no—
35#1379—31no—
36#1378—31no—
37C2×C3×#41C2×C411yescharacteristic
38C2×#235C2×C411yescharacteristic
39C3×#131—21no—
40C3×#131—21no—
41C3×#131—21no—
42C3×#131—21no—
43C2×C3×M16—21no—
44C2×C3×M16C2×C411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8C2311yescharacteristic
47#951—21no—
48#950—22no—
49#949—21no—
50#942C812yes—
51#932C812yes—
52#926C2×C411yescharacteristic
53C2×#208C2×C411yescharacteristic
54C2×#208C2×C411yescharacteristic
55#901—21no—
56#901—21no—
57#901D412yes—
58#901D412yes—
59C2×ΓL(1,𝔽25)—21no—
60C2×D3×C8—21no—
61C3×M32—21no—
62C3×M32—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C2×C3×C16D411yescharacteristic
67C2×C3×C16D411yescharacteristic
68#837—22no—
69#825—22no—
70#822—22no—
71#815—21no—
72C8×Dic3—21no—
73#812—21no—
74#812—21no—
75#812—21no—
76#812D412yes—
77C2×C3⋊C16—21no—
78C2×C3⋊C16—21no—
79#798—24no—
80D3×C16—24no—
81#575—31no—
82C2×#131—31no—
83C2×#131C2×D311yescharacteristic
84#504—62no—
85#504—62no—
86C2×M32—31no—
87C2×M32—31no—
88C2×M32—61no—
89C2×M32C2×D311yescharacteristic
90C22×C16—61no—
91C22×C16C2×D311yescharacteristic
92#444—62no—
93#443—62no—
94#430—31no—
95#421—31no—
96#421—31no—
97#405—31no—
98#350—21no—
99#350—21no—
100#350—62no—
101#349—62no—
102C3×#41—21no—
103C3×#41—21no—
104C3×#41—21no—
105C3×#41—21no—
106C2×C3×Q8C2×C811yescharacteristic
107C2×C3×D4—21no—
108C2×C3×D4C2×C811yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4C22×C411yescharacteristic
111C2×#67—21no—
112#235—21no—
113#235—21no—
114#235C22⋊C412yes—
115#235C2×C812yes—
116C2×D12M1611yescharacteristic
117C2×C4×D3—21no—
118C2×Dic6M1611yescharacteristic
119C3×M16—21no—
120C3×M16—21no—
121C3×M16—41no—
122C3×M16—41no—
123C2×C3×C8—21no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8C2×D411yescharacteristic
127C2×C3×C8C2×D411yescharacteristic
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8C2×D411yescharacteristic
130C2×C3×C8#4111yescharacteristic
131C2×C3×C8#4111yescharacteristic
132#217—21no—
133#216—22no—
134#215—22no—
135#214—22no—
136#213—22no—
137C4⋊Dic3—21no—
138C4×Dic3—21no—
139#208—21no—
140#208—21no—
141#208C22⋊C412yes—
142#208C2×C812yes—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145ΓL(1,𝔽25)—22no—
146ΓL(1,𝔽25)—22no—
147D3×C8—22no—
148D3×C8—22no—
149C3×C16—21no—
150C3×C16—21no—
151C3×C16—41no—
152C3⋊C16—22no—
153C3⋊C16—41no—
154C2×#41—31no—
155C2×#41C4×D311yescharacteristic
156#137—32no—
157#137—61no—
158AGL(1,ℤ/8)—32no—
159AGL(1,ℤ/8)—61no—
160#135—62no—
161#131—21no—
162#131—21no—
163#131—21no—
164#131—21no—
165#131—32no—
166#131—32no—
167#131—61no—
168#131—61no—
169C2×M16—21no—
170C2×M16—31no—
171C2×M16—31no—
172C2×M16—61no—
173C2×M16C4×D311yescharacteristic
174C22×C8—21no—
175C22×C8—61no—
176C22×C8C22×D311yescharacteristic
177#117—31no—
178M32—21no—
179M32—21no—
180M32—32no—
181M32—64no—
182M32—61no—
183M32—61no—
184M32—61no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—64no—
189C2×C16—61no—
190C2×C16—61no—
191C2×C16#6711yescharacteristic
192C2×C16#6711yescharacteristic
193#108—62no—
194#104—62no—
195#104—62no—
196#97—61no—
197C4×C8—61no—
198C23×C3—21no—
199C23×C3#13111yescharacteristic
200C22×D3—21no—
201C3×Q8—21no—
202C3×Q8—21no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—41no—
206C3×D4—41no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C22×C811yescharacteristic
214C2×C3×C4C2×M1611yescharacteristic
215C2×C3×C4#13111yescharacteristic
216#67—22no—
217#67—22no—
218C2×Dic3—21no—
219C2×Dic3—41no—
220D12—21no—
221D12C22⋊C812yes—
222C4×D3—22no—
223C4×D3—22no—
224Dic6—21no—
225Dic6C22⋊C812yes—
226C3×C8—41no—
227C3×C8#12011yescharacteristic
228C3×C8#12111yescharacteristic
229C3×C8#12111yescharacteristic
230C3×C8#12311yescharacteristic
231C3⋊C8—22no—
232C3⋊C8—22no—
233#41—21no—
234#41—21no—
235#41—21no—
236#41—21no—
237#41—32no—
238#41—32no—
239#41—61no—
240#41—61no—
241C2×Q8—31no—
242C2×Q8D3×C811yescharacteristic
243C2×D4—21no—
244C2×D4—31no—
245C2×D4—61no—
246C2×D4D3×C811yescharacteristic
247C22×C4—21no—
248C22×C4—61no—
249C22×C4C2×C4×D311yescharacteristic
250Q16—62no—
251SD16—62no—
252SD16—62no—
253D8—62no—
254M16—21no—
255M16—21no—
256M16—32no—
257M16—32no—
258M16—41no—
259M16—41no—
260M16—61no—
261M16—61no—
262M16—62no—
263M16—62no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8#23511yescharacteristic
272C2×C8D3×D411yescharacteristic
273C2×C8D3×D411yescharacteristic
274C2×C8D3×D411yescharacteristic
275C2×C8#23711yescharacteristic
276C2×C8C2×#6711yescharacteristic
277C4⋊C4—61no—
278C22⋊C4—61no—
279C42—61no—
280C16—21no—
281C16—21no—
282C16—41no—
283C16—62no—
284C16—121no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—81no—
288C22×C3#40811yescharacteristic
289C22×C3C8×D411yescharacteristic
290C22×C3#43511yescharacteristic
291C2×D3—22no—
292C2×D3—42no—
293C3×C4—41no—
294C3×C4#38611yescharacteristic
295C3×C4C2×C22⋊C811yescharacteristic
296C3×C4C8×D411yescharacteristic
297C3×C4#43511yescharacteristic
298Dic3—22no—
299Dic3—81no—
300C23—21no—
301C23—61no—
302C23#90711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—32no—
306Q8—61no—
307D4—21no—
308D4—21no—
309D4—32no—
310D4—41no—
311D4—41no—
312D4—62no—
313D4—62no—
314D4—61no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—121no—
322C2×C4D3×C22⋊C411yescharacteristic
323C2×C4#67⋊C411yescharacteristic
324C2×C4C2×D3×C811yescharacteristic
325C2×C4#90111yescharacteristic
326C2×C4D3×M1611yescharacteristic
327C2×C4C4×#6711yescharacteristic
328C8—41no—
329C8—62no—
330C8—62no—
331C8#88311yescharacteristic
332C8#88411yescharacteristic
333C8#93711yescharacteristic
334C8#93811yescharacteristic
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—81no—
339C2×C3#186911yescharacteristic
340D3—42no—
341C22—41no—
342C22—41no—
343C22—62no—
344C22—81no—
345C22—122no—
346C22#67⋊C811yescharacteristic
347C22#472711yescharacteristic
348C22#512711yescharacteristic
349C4—41no—
350C4—62no—
351C4—241no—
352C4D3×C22⋊C811yescharacteristic
353C4#472811yescharacteristic
354C4#495611yescharacteristic
355C4C8×#6711yescharacteristic
356C3#1298711yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—122no—
362C2#6995711yescharacteristic
subgroupG/H
centerC8#938
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC2×C3×#350C2
socleC2×C3#1869
prime radical#5822C4
cosocleC22×C3×C8C23
hypercenterC2×#350D3
hypocenterC3#12987
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×#4901C2semidirect
3. C2×C3×#350C2semidirect
4. #72007C2semidirect
5. #71438C2semidirect
6. #67813C2semidirect
7. #67805C2semidirect
33. C2×#350D3semidirect
356. C3#12987semidirect

Maximal subgroups

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

subgroupindex
#837922
C2×#49012
C2×C3×#3502
#720072
#714382
#678132
#678052
#129873