Finite Group Databasebeta

← #10694719

Subgroups — #10694719

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic
2#71970C211yesmaximal, characteristic, complemented
3#71967C211yesmaximal, characteristic, complemented
4#71686C212yesmaximal, complemented
5#71675C212yesmaximal, complemented
6#9785—31nomaximal, Sylow
7C22×C3×D8C411yescharacteristic, complemented
8C23×C3×C8C2211yescharacteristic
9C2×C3×#121C2212yes—
10C3×#466—21no—
11C3×#466—21no—
12C3×#466—21no—
13C3×#466—21no—
14C2×C3×#107C411yescharacteristic, complemented
15C2×C3×#102C412yescomplemented
16C2×#834—24no—
17#5212C2212yes—
18C4×#217C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#817—21no—
21C2×#815—21no—
22#4551—21no—
23#4550—21no—
24#4537—24no—
25C2×#466D311yescharacteristic
26#1964—32no—
27#1807—31no—
28#1752—31no—
29#1519—32no—
30C22×C3×D4—22no—
31C22×C3×D4C2×C412yes—
32C23×C3×C4C2311yescharacteristic
33C2×C3×D8—21no—
34C2×C3×D8—21no—
35C2×C3×D8—42no—
36C22×C3×C8—42no—
37C22×C3×C8C2×C411yescharacteristic
38C22×C3×C8C2×C411yescharacteristic
39C22×C3×C8D411yescharacteristic
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C3×#121—22no—
44C3×#121—22no—
45C3×#121—22no—
46C3×#121—22no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C22⋊C4—22no—
49C2×#217—22no—
50C2×#217D412yes—
51C2×C4⋊Dic3—22no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54C3×#107—21no—
55C3×#107—21no—
56C3×#102—22no—
57C3×#102—22no—
58#834—48no—
59C3⋊#95—21no—
60C3⋊#95—22no—
61#830—21no—
62#830D412yes—
63#817—42no—
64#815—42no—
65C22×D8Dic311yescharacteristic, complemented
66C23×C8C2×D311yescharacteristic
67C2×#121C2×D312yes—
68#466—21no—
69#466—21no—
70#466—21no—
71#466—21no—
72C2×#107Dic311yescharacteristic, complemented
73C2×#106—61no—
74C2×#102Dic312yescomplemented
75C2×#99—64no—
76C2×C22⋊C8—31no—
77C2×#97—61no—
78#380—32no—
79C4×C22⋊C4—31no—
80#340—61no—
81#336—61no—
82#327—64no—
83C24×C3—22no—
84C24×C3C2×D411yescharacteristic
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C2×C3×D4—44no—
92C2×C3×D4—44no—
93C22×C3×C4—22no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—22no—
103C22×Dic3—42no—
104C3×D8—42no—
105C3×D8—42no—
106C2×C3×C8—22no—
107C2×C3×C8—22no—
108C2×C3×C8—22no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C2×C3×C8C22⋊C411yescharacteristic
113C2×C3×C8C22⋊C411yescharacteristic
114C2×C3×C8C22⋊C411yescharacteristic
115C2×C3×C8C22⋊C411yescharacteristic
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120#217—24no—
121#217—44no—
122C4⋊Dic3—42no—
123C4⋊Dic3—42no—
124C4×Dic3—41no—
125C4×Dic3—41no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C22×D4—22no—
130C22×D4C2×Dic312yes—
131C23×C4C22×D311yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—42no—
135C22×C8—42no—
136C22×C8—61no—
137C22×C8C2×Dic311yescharacteristic
138C22×C8C2×Dic311yescharacteristic
139C22×C8#6711yescharacteristic
140C22×C8#6711yescharacteristic
141C22×C8#6711yescharacteristic
142C22×C8#6711yescharacteristic
143#121—22no—
144#121—22no—
145#121—22no—
146#121—22no—
147C2×C4⋊C4—62no—
148C2×C4⋊C4C2×Dic312yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—32no—
151C2×C22⋊C4—62no—
152C2×C42—61no—
153#107—21no—
154#107—21no—
155#106—122no—
156#102—22no—
157#102—22no—
158#99—128no—
159C22⋊C8—32no—
160C22⋊C8—61no—
161#97—122no—
162#95—61no—
163#95—62no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—42no—
167C23×C3—44no—
168C23×C3—82no—
169C23×C3—82no—
170C23×C3C4×D411yescharacteristic
171C23×C3#12111yescharacteristic
172C23×C3#12711yescharacteristic
173C23×C3AGL(1,ℤ/8)14yes—
174C3×D4—42no—
175C3×D4—42no—
176C3×D4—82no—
177C3×D4—84no—
178C3×D4—82no—
179C3×D4—82no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—44no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—82no—
199C2×Dic3—82no—
200C2×Dic3—82no—
201C3×C8—21no—
202C3×C8—21no—
203C3×C8—21no—
204C3×C8#9912yescomplemented
205C3⋊C8—42no—
206C24—22no—
207C24D3×D411yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—62no—
220C22×C4—122no—
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223C22×C4#23711yescharacteristic
224C22×C4C22×Dic311yescharacteristic
225C22×C4C2×#6711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227D8—42no—
228D8—42no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—62no—
236C2×C8—121no—
237C2×C8—121no—
238C2×C8#21711yescharacteristic
239C2×C8#21711yescharacteristic
240C2×C8#21711yescharacteristic
241C2×C8#21711yescharacteristic
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—122no—
245C4⋊C4—122no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—64no—
249C22⋊C4—124no—
250C42—121no—
251C42—121no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—44no—
259C22×C3—82no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—164no—
263C22×C3#39011yescharacteristic
264C22×C3#44212yes—
265C22×C3#46912yes—
266C22×C3#49612yes—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—82no—
271C3×C4C2×#9911yescharacteristic
272C3×C4#41011yescharacteristic
273Dic3—82no—
274Dic3—162no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—44no—
279C23—82no—
280C23—82no—
281C23#91114yes—
282C23D4×Dic311yescharacteristic
283C23#93811yescharacteristic
284C23#94011yescharacteristic
285D4—42no—
286D4—42no—
287D4—82no—
288D4—82no—
289D4—82no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—242no—
306C2×C4—242no—
307C2×C4—242no—
308C2×C4D4×Dic311yescharacteristic
309C2×C4#93611yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4C2×#21711yescharacteristic
312C8—21no—
313C8—21no—
314C8—21no—
315C8—122no—
316C8#83412yescomplemented
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—22no—
320C2×C3—162no—
321C2×C3#196411yescharacteristic
322C2×C3#197211yescharacteristic
323C2×C3#197611yescharacteristic
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—164no—
335C22#515212yes—
336C22#515412yes—
337C22#515912yes—
338C22#522111yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—242no—
344C4—482no—
345C4C2×#83411yescharacteristic
346C4#524011yescharacteristic
347C3#978511yescharacteristic, complemented, Sylow
348C2—22no—
349C2—22no—
350C2—22no—
351C2—162no—
352C2#7168611yescharacteristic
353C2#7203911yescharacteristic
354C2#7208611yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#466C2
socleC22×C3#390
cosocleC23×C3×C4C23
hypercenterC2×#466D3
hypocenterC3#9785
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. #71970C2semidirect
3. #71967C2semidirect
4. #71686C2semidirect
5. #71675C2semidirect
7. C22×C3×D8C4semidirect
14. C2×C3×#107C4semidirect
15. C2×C3×#102C4semidirect
65. C22×D8Dic3semidirect
72. C2×#107Dic3semidirect
74. C2×#102Dic3semidirect
204. C3×C8#99semidirect
316. C8#834semidirect
347. C3#9785semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
#719702
#719672
#716862
#716752
#97853