Finite Group Databasebeta

← #10795607

Subgroups — #10795607

Order 768.

1086 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4880C212yesmaximal
2C2×#4860C212yesmaximal
3C2×D3×#97C211yesmaximal, characteristic
4#74318C211yesmaximal, characteristic
5#74148C211yesmaximal, characteristic
6C3×#1847C211yesmaximal, characteristic, complemented
7#70608C212yesmaximal, complemented
8#70556C212yesmaximal, complemented
9#69816C211yesmaximal, characteristic, complemented
10#69578C211yesmaximal, characteristic, complemented
11#69550C211yesmaximal, characteristic, complemented
12#20799—31nomaximal, Sylow
13C22×D3×C8C2212yes—
14C2×#892C2211yescharacteristic
15C2×#892C2212yes—
16C2×#892C2211yescharacteristic
17C2×C42×D3C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×#97C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C3×#382C2211yescharacteristic
23C2×#818C2212yes—
24C2×#817C2212yes—
25C2×#815C2212yes—
26#4987—22no—
27#4977C2212yes—
28#4975C2211yescharacteristic
29#4975C2211yescharacteristic
30Dic3×C4⋊C4—22no—
31C2×#808C2212yes—
32C2×#808C2212yes—
33C2×#807C2212yes—
34C2×#807C2212yes—
35#4936C2211yescharacteristic
36#4935C2211yescharacteristic
37C2×#803C2211yescharacteristic
38#4880—28no—
39#4860—28no—
40D3×#97C418yescomplemented
41#4667—22no—
42#4649—22no—
43C2×#427—32no—
44C2×#427—32no—
45C22×#97—31no—
46#2285—31no—
47#2285—31no—
48#1847—32no—
49#1847—32no—
50#1847—31no—
51#1847—31no—
52#1847—31no—
53#1847D311yescharacteristic, complemented
54C22×C4×D3D412yes—
55C22×C4×D3C2311yescharacteristic
56C22×C3×C8C2312yes—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4⋊Dic3C2311yescharacteristic
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3D412yes—
75C2×C4×Dic3C2311yescharacteristic
76C22×C3⋊C8C2312yes—
77C2×D3×C8C2×C418yes—
78C2×D3×C8D418yes—
79C2×D3×C8Q818yes—
80#892—24no—
81#892—24no—
82#892—24no—
83#892—24no—
84C42×D3C2×C414yes—
85C42×D3Q814yes—
86C3×#107—24no—
87C3×#106—24no—
88C3×#97C2×C414yes—
89C3×#95—22no—
90C3⋊#95—22no—
91C3⋊#95—24no—
92#818—24no—
93#817—24no—
94#815C2×C418yes—
95#808—24no—
96#808—24no—
97#807—24no—
98#807—24no—
99#803C2×C414yes—
100C23×C8—32no—
101C2×#117—32no—
102C2×#117—31no—
103C2×#117—31no—
104C22×C42—31no—
105#427—68no—
106#427—68no—
107C2×#107—32no—
108C2×#107—32no—
109C2×#107—32no—
110C2×#107C2×D312yes—
111C2×#106—32no—
112C2×#106—32no—
113C2×#106—32no—
114C2×#106C2×D312yes—
115C2×#97—32no—
116C2×#97—31no—
117C2×#97—38no—
118C2×#97C2×D311yescharacteristic
119#384—31no—
120#384—31no—
121#384—31no—
122#384C2×D311yescharacteristic
123#382—32no—
124#382—31no—
125#382C2×D311yescharacteristic
126#380—62no—
127#380—62no—
128C4×C4⋊C4—62no—
129C4×C4⋊C4—62no—
130C23×D3C2×D411yescharacteristic
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—22no—
139C22×Dic3C2×D412yes—
140C22×Dic3C2×D411yescharacteristic
141C2×C4×D3—42no—
142C2×C4×D3—42no—
143C2×C4×D3C4⋊C418yes—
144C2×C4×D3C22×C412yes—
145C2×C4×D3C2×D414yes—
146C2×C4×D3C2×Q814yes—
147C2×C4×D3C2×Q812yes—
148C2×C3×C8C22×C414yes—
149C2×C3×C8C2×D414yes—
150C2×C3×C8C2×Q814yes—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—44no—
156C3×C42C22×C412yes—
157C3×C42C2×Q812yes—
158(C2×D3)⋊C4—42no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—42no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—44no—
166Dic3⋊C4—44no—
167Dic3⋊C4—44no—
168C4×Dic3—42no—
169C4×Dic3—44no—
170C4×Dic3—42no—
171C4×Dic3C4⋊C418yes—
172C4×Dic3C22×C412yes—
173C4×Dic3C2×Q812yes—
174C2×C3⋊C8C22×C414yes—
175C2×C3⋊C8C2×D414yes—
176C2×C3⋊C8C2×Q814yes—
177D3×C8C4⋊C4132yescomplemented
178C23×C4—31no—
179C23×C4—32no—
180C22×C8—32no—
181C22×C8—38no—
182C22×C8—38no—
183C22×C8—38no—
184C22×C8C22×D312yes—
185#117—64no—
186#117—64no—
187#117—64no—
188#117—64no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4C22×D312yes—
197C2×C4⋊C4C22×D311yescharacteristic
198C2×C4⋊C4C22×D311yescharacteristic
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—62no—
202C2×C42—32no—
203C2×C42—31no—
204C2×C42—34no—
205C2×C42—34no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—62no—
209C2×C42C22×D311yescharacteristic
210#107—24no—
211#107—64no—
212#107—64no—
213#107—64no—
214#106—24no—
215#106—64no—
216#106—64no—
217#106—64no—
218#97—38no—
219#97—34no—
220#97C4×D314yes—
221#95—22no—
222#95—64no—
223#95—62no—
224C23×C3C22×D411yescharacteristic
225C22×D3—22no—
226C22×D3—22no—
227C22×D3C2×C4⋊C412yes—
228C22×D3#12712yes—
229C22×D3C4⋊Q812yes—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—44no—
238C2×C3×C4C2×C4⋊C414yes—
239C2×C3×C4C23×C411yescharacteristic
240C2×C3×C4C22×D412yes—
241C2×C3×C4C22×Q812yes—
242C2×C3×C4C22×Q811yescharacteristic
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—44no—
251C2×Dic3C2×C4⋊C414yes—
252C2×Dic3C2×C4⋊C412yes—
253C2×Dic3#12712yes—
254C2×Dic3C4⋊Q812yes—
255C4×D3—48no—
256C4×D3C2×C4⋊C418yes—
257C4×D3C4⋊Q818yes—
258C3×C8C2×C4⋊C418yes—
259C3⋊C8C2×C4⋊C418yes—
260C24—31no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—34no—
267C22×C4—32no—
268C22×C4—34no—
269C22×C4—32no—
270C22×C4—38no—
271C22×C4—61no—
272C22×C4—62no—
273C22×C4—61no—
274C22×C4—122no—
275C22×C4—122no—
276C22×C4D3×D412yes—
277C22×C4C23×D311yescharacteristic
278C2×C8—34no—
279C2×C8—34no—
280C2×C8—34no—
281C2×C8—332no—
282C2×C8C2×C4×D314yes—
283C2×C8D3×D414yes—
284C2×C8D3×Q814yes—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—44no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—124no—
295C4⋊C4—124no—
296C4⋊C4—124no—
297C22⋊C4—124no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C42—38no—
301C42—32no—
302C42—32no—
303C42—124no—
304C42—122no—
305C42—122no—
306C42C2×C4×D312yes—
307C42D3×Q812yes—
308C22×C3C22×C4⋊C411yescharacteristic
309C22×C3C2×#12711yescharacteristic
310C22×C3C2×C4⋊Q811yescharacteristic
311C22×C3#57512yes—
312C22×C3#57512yes—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—24no—
316C2×D3—24no—
317C2×D3#38414yes—
318C3×C4—24no—
319C3×C4—81no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4C22×C4⋊C412yes—
323C3×C4C2×C4⋊Q812yes—
324Dic3—24no—
325Dic3—81no—
326Dic3—81no—
327Dic3—82no—
328Dic3#38414yes—
329C23—32no—
330C23—32no—
331C23—32no—
332C23—62no—
333C23—62no—
334C23C2×D3×D411yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—38no—
341C2×C4—38no—
342C2×C4—34no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—128no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—124no—
357C2×C4—122no—
358C2×C4D3×C4⋊C414yes—
359C2×C4C22×C4×D311yescharacteristic
360C2×C4C2×D3×D412yes—
361C2×C4C2×D3×Q811yescharacteristic
362C2×C4C2×D3×Q812yes—
363C8—38no—
364C8D3×C4⋊C418yes—
365C2×C3C2×#38411yescharacteristic
366C2×C3#291112yes—
367C2×C3#315011yescharacteristic
368C2×C3#315011yescharacteristic
369C2×C3#316212yes—
370D3—24no—
371C22—34no—
372C22—64no—
373C22—64no—
374C22—62no—
375C22—62no—
376C22C2×D3×C4⋊C411yescharacteristic
377C22D3×#12711yescharacteristic
378C22D3×C4⋊Q811yescharacteristic
379C22#577312yes—
380C22#577712yes—
381C4—24no—
382C4—34no—
383C4—64no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—241no—
388C4—242no—
389C4—241no—
390C4C2×D3×C4⋊C412yes—
391C4D3×C4⋊Q812yes—
392C3#2079911yescharacteristic, complemented, Sylow
393C2—64no—
394C2D3×#38411yescharacteristic
395C2#7920812yes—
396C2#8184411yescharacteristic
397C2#8186911yescharacteristic
398C2#8197312yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1847C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1847D3
hypocenterC3#20799
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
6. C3×#1847C2semidirect
7. #70608C2semidirect
8. #70556C2semidirect
9. #69816C2semidirect
10. #69578C2semidirect
11. #69550C2semidirect
40. D3×#97C4semidirect
53. #1847D3semidirect
177. D3×C8C4⋊C4semidirect
392. C3#20799semidirect

Maximal subgroups

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

subgroupindex
C2×#48802
C2×#48602
C2×D3×#972
#743182
#741482
C3×#18472
#706082
#705562
#698162
#695782
#695502
#207993