Finite Group Databasebeta

← #10712078

Subgroups — #10712078

Order 768.

508 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#83794C211yesmaximal, characteristic
2C22×D3×C16C211yesmaximal, characteristic, complemented
3C2×C3×#350C211yesmaximal, characteristic, complemented
4C2×#4561C211yesmaximal, characteristic, complemented
5#71431C211yesmaximal, characteristic, complemented
6#67815C211yesmaximal, characteristic, complemented
7#67805C211yesmaximal, characteristic, complemented
8#12983—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5828C411yescharacteristic
11C22×D3×C8C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14#5234C411yescharacteristic
15#5127—22no—
16#5126C411yescharacteristic
17#5119C411yescharacteristic
18C8×#67—22no—
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C22×C3⋊C16C2211yescharacteristic
22C2×D3×C16—22no—
23C2×D3×C16—22no—
24C2×D3×C16—22no—
25C3×#350—22no—
26#4561—22no—
27(C2×D3)⋊C16—22no—
28C23×C16—31no—
29#2929—31no—
30C2×#350—31no—
31C2×#350D311yescharacteristic, complemented
32#2112—31no—
33#1379—31no—
34#1379—31no—
35C2×C3×#41C2×C411yescharacteristic
36C22×C4×D3C2×C411yescharacteristic
37C3×#131—21no—
38C3×#131—21no—
39C3×#131—21no—
40C3×#131—21no—
41C2×C3×M16—21no—
42C2×C3×M16C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8C2311yescharacteristic
45#946C812yes—
46#938C812yes—
47#937—21no—
48#935—21no—
49C4×#67—22no—
50#926C2×C411yescharacteristic
51C2×#208C2×C411yescharacteristic
52C22×C3⋊C8C2×C411yescharacteristic
53C2×D3×C8—21no—
54C2×D3×C8—21no—
55C2×D3×C8—21no—
56C2×D3×C8—21no—
57C2×D3×C8D412yes—
58C2×D3×C8D412yes—
59C3×M32—21no—
60C3×M32—21no—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16D412yescomplemented
65#830—22no—
66(C2×D3)⋊C8—22no—
67#815—21no—
68Dic3⋊C8—22no—
69C8×Dic3—21no—
70#812—21no—
71#812—21no—
72C2×C3⋊C16—21no—
73C2×C3⋊C16—21no—
74C2×C3⋊C16—21no—
75C2×C3⋊C16D412yescomplemented
76D3×C16—28no—
77C2×#131C2×D311yescharacteristic
78C23×C8—31no—
79#525—31no—
80C2×M32—31no—
81C2×M32C2×D311yescharacteristic
82C22×C16—31no—
83C22×C16—62no—
84C22×C16—62no—
85C22×C16—62no—
86C22×C16C2×D311yescharacteristic
87#435—62no—
88C8×D4—62no—
89#424—31no—
90#408—31no—
91#407—31no—
92#405—31no—
93#350—22no—
94#350—62no—
95C22⋊C16—62no—
96C23×D3M1611yescharacteristic
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—21no—
100C3×#41—21no—
101C2×C3×Q8C2×C811yescharacteristic
102C2×C3×D4—21no—
103C2×C3×D4C2×C811yescharacteristic
104C22×C3×C4—21no—
105C22×C3×C4C22×C411yescharacteristic
106C2×#67—22no—
107C22×Dic3M1611yescharacteristic
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3C22⋊C412yes—
113C2×C4×D3C2×C812yes—
114C3×M16—21no—
115C3×M16—21no—
116C3×M16—41no—
117C3×M16—41no—
118C2×C3×C8—21no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8C2×D411yescharacteristic
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8#4111yescharacteristic
126C2×C3×C8#4111yescharacteristic
127#217—21no—
128#217—22no—
129(C2×D3)⋊C4—22no—
130C4⋊Dic3C2×C812yes—
131Dic3⋊C4—22no—
132C4×Dic3—21no—
133#208—21no—
134#208—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8C22⋊C412yes—
139D3×C8—22no—
140D3×C8—22no—
141D3×C8—22no—
142D3×C8—22no—
143C3×C16—22no—
144C3×C16—41no—
145C3⋊C16—22no—
146C3⋊C16—41no—
147C2×#41C4×D311yescharacteristic
148C23×C4—31no—
149#131—21no—
150#131—21no—
151#131—21no—
152#131—21no—
153C2×M16—21no—
154C2×M16—31no—
155C2×M16C4×D311yescharacteristic
156C22×C8—21no—
157C22×C8—31no—
158C22×C8—32no—
159C22×C8—32no—
160C22×C8—61no—
161C22×C8—61no—
162C22×C8—61no—
163C22×C8—61no—
164C22×C8C22×D311yescharacteristic
165#123—61no—
166C22⋊Q8—32no—
167#121—32no—
168#120—61no—
169C4×D4—62no—
170#117—31no—
171M32—21no—
172M32—21no—
173M32—61no—
174M32—61no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—32no—
179C2×C16—68no—
180C2×C16—61no—
181C2×C16—61no—
182C2×C16—61no—
183C2×C16#6712yescomplemented
184C4⋊C8—62no—
185C22⋊C8—62no—
186C22⋊C8—62no—
187#97—61no—
188C4×C8—61no—
189C23×C3—21no—
190C23×C3C2×M1611yescharacteristic
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—24no—
194C22×D3C22⋊C812yes—
195C3×Q8—21no—
196C3×Q8—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—41no—
200C3×D4—41no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C22×C811yescharacteristic
208C2×C3×C4#13111yescharacteristic
209C2×C3×C4#13111yescharacteristic
210#67—44no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—22no—
214C2×Dic3C22⋊C812yes—
215C4×D3—22no—
216C4×D3—22no—
217C4×D3—22no—
218C4×D3—22no—
219C3×C8—41no—
220C3×C8#12011yescharacteristic
221C3×C8#12111yescharacteristic
222C3×C8#12111yescharacteristic
223C3×C8#12311yescharacteristic
224C3⋊C8—22no—
225C3⋊C8—41no—
226C24—31no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231C2×Q8D3×C811yescharacteristic
232C2×D4—21no—
233C2×D4—62no—
234C2×D4D3×C811yescharacteristic
235C22×C4—21no—
236C22×C4—31no—
237C22×C4—32no—
238C22×C4—32no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4C2×C4×D311yescharacteristic
244M16—21no—
245M16—21no—
246M16—41no—
247M16—41no—
248M16—61no—
249M16—61no—
250C2×C8—21no—
251C2×C8—32no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—62no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—61no—
261C2×C8#23511yescharacteristic
262C2×C8D3×D411yescharacteristic
263C2×C8D3×D411yescharacteristic
264C2×C8D3×D411yescharacteristic
265C2×C8#23711yescharacteristic
266C2×C8C2×#6711yescharacteristic
267C4⋊C4—32no—
268C4⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—61no—
272C42—61no—
273C16—22no—
274C16—41no—
275C16—62no—
276C16—121no—
277C22×C3—22no—
278C22×C3—42no—
279C22×C3—41no—
280C22×C3C2×C22⋊C811yescharacteristic
281C22×C3#43511yescharacteristic
282C22×C3#43511yescharacteristic
283C2×D3—24no—
284C2×D3—24no—
285C2×D3—22no—
286C2×D3—22no—
287C2×D3#35014yescomplemented
288C3×C4—41no—
289C3×C4#38611yescharacteristic
290C3×C4#40811yescharacteristic
291C3×C4C8×D411yescharacteristic
292C3×C4C8×D411yescharacteristic
293Dic3—22no—
294Dic3—42no—
295C23—21no—
296C23—32no—
297C23—64no—
298C23—61no—
299C23—61no—
300C23D3×M1611yescharacteristic
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—41no—
306D4—41no—
307D4—124no—
308C2×C4—21no—
309C2×C4—32no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4D3×C22⋊C411yescharacteristic
320C2×C4#67⋊C411yescharacteristic
321C2×C4C2×D3×C811yescharacteristic
322C2×C4#90111yescharacteristic
323C2×C4#90711yescharacteristic
324C2×C4C4×#6711yescharacteristic
325C8—41no—
326C8—62no—
327C8—121no—
328C8#88311yescharacteristic
329C8#88411yescharacteristic
330C8#93711yescharacteristic
331C8#93811yescharacteristic
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—81no—
335C2×C3#186911yescharacteristic
336C2×C3C2×#35012yes—
337D3—24no—
338C22—22no—
339C22—34no—
340C22—42no—
341C22—41no—
342C22—64no—
343C22—64no—
344C22—62no—
345C22—62no—
346C22D3×C22⋊C811yescharacteristic
347C22#472811yescharacteristic
348C22#512711yescharacteristic
349C4—41no—
350C4—62no—
351C4—122no—
352C4#67⋊C811yescharacteristic
353C4#472711yescharacteristic
354C4#495611yescharacteristic
355C4C8×#6711yescharacteristic
356C3#1298311yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—64no—
360C2—81no—
361C2D3×#35012yes—
362C2#6995711yescharacteristic
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC2×C3×#350C2
socleC22×C3C2×C22⋊C8
prime radical#5828C4
cosocleC22×C3×C8C23
hypercenterC2×#350D3
hypocenterC3#12983
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C22×D3×C16C2semidirect
3. C2×C3×#350C2semidirect
4. C2×#4561C2semidirect
5. #71431C2semidirect
6. #67815C2semidirect
7. #67805C2semidirect
31. C2×#350D3semidirect
64. C2×C3×C16D4semidirect
75. C2×C3⋊C16D4semidirect
183. C2×C16#67semidirect
287. C2×D3#350semidirect
356. C3#12983semidirect

Maximal subgroups

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

subgroupindex
#837942
C22×D3×C162
C2×C3×#3502
C2×#45612
#714312
#678152
#678052
#129833