Finite Group Databasebeta

← #10896515

Subgroups — #10896515

Order 768.

684 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C4⋊Q8C211yesmaximal, characteristic
2#75966C211yesmaximal, characteristic
3C3×#2057C211yesmaximal, characteristic
4#71783C211yesmaximal, characteristic
5#70616C212yesmaximal
6#70609C212yesmaximal
7#69612C211yesmaximal, characteristic
8#68817C212yesmaximal
9#68802C212yesmaximal
10#68643C211yesmaximal, characteristic
11#68605C211yesmaximal, characteristic
12#29278—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15C2×#890C2211yescharacteristic
16C2×C3×#107C2212yes—
17C3×#389C2211yescharacteristic
18#5229—21no—
19C2×Dic3⋊C8C2212yes—
20#4984—21no—
21#4984C2212yes—
22#4978—21no—
23#4977C2211yescharacteristic
24Dic3×C4⋊C4—21no—
25Dic3×C4⋊C4—21no—
26Dic3×C4⋊C4C2212yes—
27Dic3×C4⋊C4C2211yescharacteristic
28Dic3×C4⋊C4C2211yescharacteristic
29C2×#808C2212yes—
30#4935C2211yescharacteristic
31C42×Dic3C2211yescharacteristic
32#4928C2211yescharacteristic
33#4653—21no—
34#4649—21no—
35C3×#340C2212yes—
36C3×#336C2211yescharacteristic
37#4551C2212yes—
38#4551C2212yes—
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C4⋊Q8—31no—
45C4×#125—31no—
46#2100—31no—
47#2100—31no—
48#2058—31no—
49#2057D311yescharacteristic
50#1989—32no—
51#1949—32no—
52#1919—32no—
53#1918—32no—
54#1831—31no—
55C22×C3×Q8—21no—
56C22×C3×C8C2312yes—
57C3×C4⋊Q8—24no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42C2311yescharacteristic
64Q8×Dic3—44no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3C2311yescharacteristic
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3D411yescharacteristic
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3C2311yescharacteristic
82C2×C4×Dic3C2311yescharacteristic
83C22×C3⋊C8C2312yes—
84#890—22no—
85#890Q814yescomplemented
86C3×#107—24no—
87C3×#95—21no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95Dic3⋊C8—24no—
96#808—24no—
97C2×C4⋊Q8C2×D311yescharacteristic
98C2×#125—31no—
99C2×C4×Q8—61no—
100C2×#107C2×D312yes—
101C2×#106—32no—
102C2×C4⋊C8—32no—
103#391—61no—
104#389—32no—
105#389—31no—
106#389—61no—
107#389C2×D311yescharacteristic
108#387—61no—
109#387—61no—
110#384—31no—
111#382—31no—
112C4×C4⋊C4—31no—
113C4×C4⋊C4—31no—
114C4×C4⋊C4—32no—
115C4×C4⋊C4—61no—
116C4×C4⋊C4—61no—
117C4×C4⋊C4—61no—
118C43—31no—
119#340—32no—
120#340—32no—
121#340—32no—
122#340—32no—
123#340—32no—
124#340C2×D312yes—
125#336—32no—
126#336—31no—
127#336C2×D311yescharacteristic
128C2×C3×Q8—24no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3C2×D411yescharacteristic
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D411yescharacteristic
143C22×Dic3C2×D411yescharacteristic
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4C2×Q812yes—
153C3×C4⋊C4#4114yes—
154C3×C42—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—24no—
157C4⋊Dic3—22no—
158C4⋊Dic3—24no—
159C4⋊Dic3—42no—
160C4⋊Dic3—42no—
161C4⋊Dic3—44no—
162C4⋊Dic3C2×Q812yes—
163Dic3⋊C4—24no—
164Dic3⋊C4—42no—
165C4×Dic3—22no—
166C4×Dic3—22no—
167C4×Dic3—22no—
168C4×Dic3—22no—
169C4×Dic3—41no—
170C4×Dic3—42no—
171C4×Dic3—44no—
172C4×Dic3—42no—
173C4×Dic3C2×Q812yes—
174C4×Dic3#4112yes—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C22×Q8—21no—
179C22×C8—32no—
180C22×C8C22×D312yes—
181C4⋊Q8—24no—
182#125—34no—
183#125—62no—
184C4×Q8—124no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C22×D311yescharacteristic
197C2×C4⋊C4C22×D312yes—
198C2×C4⋊C4C22×D311yescharacteristic
199C2×C42—31no—
200C2×C42—31no—
201C2×C42—31no—
202C2×C42—31no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—62no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42C22×D311yescharacteristic
209#107—24no—
210#106—64no—
211C4⋊C8—64no—
212#95—21no—
213#95—61no—
214#95—61no—
215#95—62no—
216#95—61no—
217#95—61no—
218#95—62no—
219#95—61no—
220C23×C3C22×D411yescharacteristic
221C3×Q8—48no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4#13512yes—
236C2×C3×C4#13712yes—
237C2×C3×C4C22×Q811yescharacteristic
238C2×C3×C4C2×#4112yes—
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C42−1+412yes—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—22no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3C22⋊Q812yes—
260C2×Dic3C22⋊Q812yes—
261C2×Dic3AGL(1,ℤ/8)12yes—
262C3×C8—42no—
263C3⋊C8—42no—
264C2×Q8—24no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—62no—
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4C23×D311yescharacteristic
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—32no—
291C4⋊C4—42no—
292C4⋊C4—64no—
293C4⋊C4—64no—
294C4⋊C4—61no—
295C4⋊C4—62no—
296C4⋊C4—64no—
297C4⋊C4—122no—
298C4⋊C4—122no—
299C4⋊C4—124no—
300C4⋊C4—122no—
301C4⋊C4#23714yes—
302C4⋊C4D3×Q812yes—
303C42—22no—
304C42—32no—
305C42—32no—
306C42—62no—
307C42—62no—
308C42—62no—
309C42—62no—
310C42—121no—
311C42—122no—
312C42—124no—
313C42—122no—
314C22×C3C2×C22⋊Q811yescharacteristic
315C22×C3#52511yescharacteristic
316C22×C3#53611yescharacteristic
317C22×C3C2×#13511yescharacteristic
318C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C3C2×#13711yescharacteristic
320C22×C3#57511yescharacteristic
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—44no—
325C3×C4—81no—
326C3×C4#55612yes—
327C3×C4#55712yes—
328Dic3—22no—
329Dic3—22no—
330Dic3—22no—
331Dic3—42no—
332Dic3—41no—
333Dic3—81no—
334Dic3—82no—
335C23C2×D3×D411yescharacteristic
336Q8—48no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4#91212yes—
372C2×C4#91812yes—
373C2×C4C2×#23712yes—
374C2×C4C2×D3×Q811yescharacteristic
375C2×C4#100712yes—
376C2×C4D3×#4111yescharacteristic
377C8—42no—
378C8—122no—
379C2×C3#258111yescharacteristic
380C2×C3#307611yescharacteristic
381C2×C3#307911yescharacteristic
382C2×C3#310511yescharacteristic
383C2×C3#311711yescharacteristic
384C2×C3#323111yescharacteristic
385C2×C3#323411yescharacteristic
386C22#558611yescharacteristic
387C22D3×C22⋊Q811yescharacteristic
388C22#572411yescharacteristic
389C22C2×#91211yescharacteristic
390C22C2×#91811yescharacteristic
391C22D3×AGL(1,ℤ/8)11yescharacteristic
392C22#577711yescharacteristic
393C4—22no—
394C4—22no—
395C4—42no—
396C4—44no—
397C4—62no—
398C4—62no—
399C4—62no—
400C4—81no—
401C4—121no—
402C4—122no—
403C4—241no—
404C4—242no—
405C4#564112yes—
406C4#572112yes—
407C3#2927811yescharacteristic, complemented, Sylow
408C2#7719111yescharacteristic
409C2#7998911yescharacteristic
410C2#8017511yescharacteristic
411C2#8096411yescharacteristic
412C2#8110911yescharacteristic
413C2#8194411yescharacteristic
414C2#8197811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socle, prime radicalC23×C3C22×D4
hypercenter#2057D3
hypocenterC3#29278
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
85. #890Q8semidirect
407. C3#29278semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊Q82
#759662
C3×#20572
#717832
#706162
#706092
#696122
#688172
#688022
#686432
#686052
#292783