Finite Group Databasebeta

← #10696400

Subgroups — #10696400

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4841C211yesmaximal, characteristic
2C2×#4680C211yesmaximal, characteristic, complemented
3#73800C211yesmaximal, characteristic
4C2×#4486C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic
6#69490C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10068—31nomaximal, Sylow
9C22×#204—21no—
10C2×#897—22no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17C2×#817—22no—
18C2×#816—21no—
19#4939—21no—
20#4935—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4841—22no—
23#4841—22no—
24#4680—22no—
25#4680—22no—
26#4677—22no—
27#4529—22no—
28#4486—22no—
29#4459—22no—
30C2×#498—31no—
31C2×#461—31no—
32#2542—31no—
33#1988—31no—
34C2×#332—31no—
35C2×D4⋊C8—31no—
36#1279D311yescharacteristic, complemented
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×Dic6—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—24no—
47C2×(C2×D3)⋊C4—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—24no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D412yes—
53C2×C4⋊Dic3D411yescharacteristic
54C22×C3⋊C8—21no—
55C2×#204—24no—
56C2×#204—42no—
57C2×#204—42no—
58#897—44no—
59#897—44no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71#869D412yes—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8D412yes—
76C3×#95—21no—
77#817—28no—
78#816—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C22×SD16—61no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8—31no—
84C2×C22⋊Q8—62no—
85C2×C4×D4—31no—
86#498—62no—
87#498—62no—
88#461—62no—
89#461—62no—
90C2×#106—62no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#103—61no—
94C2×C4×C8C2×D311yescharacteristic
95#399—62no—
96#390—61no—
97#384—61no—
98C4×C4⋊C4C2×D311yescharacteristic
99#340—62no—
100#332—62no—
101D4⋊C8—62no—
102C23×D3—21no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—22no—
109C22×Dic3—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12SD1612yes—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—42no—
119C2×C4×D3—42no—
120C2×Dic6—22no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×C3×C8—24no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4SD1614yescomplemented
137C3×C42—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—48no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—48no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—42no—
159#204—48no—
160C22×Q8—61no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—64no—
165C2×SD16—122no—
166C2×SD16—122no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—32no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C22⋊Q8—124no—
179C22⋊Q8—124no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4D1212yes—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#106—68no—
196C4⋊C8—21no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8D1212yes—
200#103—64no—
201C4×C8—21no—
202C4×C8D1212yes—
203#95—21no—
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—42no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×C3×C4C2×SD1612yes—
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)12yes—
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—44no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233D12—22no—
234D12—22no—
235D12—41no—
236D12—41no—
237C4×D3—82no—
238C4×D3—81no—
239C4×D3—81no—
240Dic6—42no—
241Dic6—42no—
242Dic6—82no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—24no—
248C3×C8—42no—
249C3⋊C8—81no—
250C24—61no—
251C2×Q8—62no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—122no—
256C2×Q8—122no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×Q8—121no—
260C2×Q8—121no—
261C2×D4—32no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—22no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4C2×D1211yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278SD16—128no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—121no—
285C2×C8—122no—
286C4⋊C4—22no—
287C4⋊C4—61no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—61no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—128no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4#20414yescomplemented
299C22⋊C4—128no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—41no—
303C42—41no—
304C42C2×D1211yescharacteristic
305C42C2×D1211yescharacteristic
306C42D3×D411yescharacteristic
307C42#23911yescharacteristic
308C22×C3#39211yescharacteristic
309C22×C3#44811yescharacteristic
310C22×C3#44911yescharacteristic
311C22×C3#45012yes—
312C22×C3#45612yes—
313C2×D3—44no—
314C2×D3—42no—
315C2×D3—82no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4#45011yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#46012yes—
324C3×C4#46511yescharacteristic
325C3×C4#46811yescharacteristic
326C3×C4#49211yescharacteristic
327C3×C4#49611yescharacteristic
328Dic3—82no—
329Dic3—82no—
330Dic3—81no—
331C23—62no—
332C23—121no—
333C23—122no—
334C23—121no—
335C23—122no—
336C23#88211yescharacteristic
337Q8—122no—
338Q8—122no—
339Q8—241no—
340Q8—241no—
341Q8—241no—
342Q8—241no—
343Q8—242no—
344D4—62no—
345D4—62no—
346D4—121no—
347D4—121no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—124no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—242no—
362C2×C4#87411yescharacteristic
363C2×C4#88211yescharacteristic
364C2×C4#88211yescharacteristic
365C2×C4#89511yescharacteristic
366C2×C4#89511yescharacteristic
367C2×C4#89511yescharacteristic
368C2×C4C2×#20412yes—
369C2×C4#90812yes—
370C2×C4#91111yescharacteristic
371C2×C4D3×SD1611yescharacteristic
372C2×C4#91611yescharacteristic
373C2×C4#91811yescharacteristic
374C8—24no—
375C8—42no—
376C8—241no—
377C2×C3#165112yes—
378C2×C3#165412yes—
379C2×C3#200011yescharacteristic
380C2×C3#200511yescharacteristic
381C2×C3#201411yescharacteristic
382D3—82no—
383C22—122no—
384C22—124no—
385C22—241no—
386C22—242no—
387C22—241no—
388C22#467111yescharacteristic
389C22#473112yes—
390C22#478111yescharacteristic
391C22#480611yescharacteristic
392C22#482612yes—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—242no—
397C4—241no—
398C4—242no—
399C4#469311yescharacteristic
400C4#471211yescharacteristic
401C4#477611yescharacteristic
402C4#480811yescharacteristic
403C4#484812yes—
404C4#486411yescharacteristic
405C4#488611yescharacteristic
406C3#1006811yescharacteristic, complemented, Sylow
407C2—242no—
408C2#6818512yes—
409C2#6825112yes—
410C2#6867611yescharacteristic
411C2#6876811yescharacteristic
412C2#6888711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1279D3
hypocenterC3#10068
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. C2×#4680C2semidirect
4. C2×#4486C2semidirect
6. #69490C2semidirect
7. C3×#1279C2semidirect
36. #1279D3semidirect
136. C3×C4⋊C4SD16semidirect
298. C4⋊C4#204semidirect
406. C3#10068semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
C2×#46802
#738002
C2×#44862
C2×#44592
#694902
C3×#12792
#100683