Finite Group Databasebeta

← #10896461

Subgroups — #10896461

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C4⋊Q8C211yesmaximal, characteristic
2#75967C211yesmaximal, characteristic
3C3×#2057C211yesmaximal, characteristic
4#71782C211yesmaximal, characteristic
5#70615C212yesmaximal
6#70609C212yesmaximal
7#69611C211yesmaximal, characteristic
8#68815C212yesmaximal
9#68800C212yesmaximal
10#68642C211yesmaximal, characteristic
11#68605C211yesmaximal, characteristic
12#29272—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15C2×#888C2211yescharacteristic
16C2×#887—21no—
17C2×C3×#107C2212yes—
18C3×#389C2211yescharacteristic
19#5229—21no—
20C2×Dic3⋊C8C2212yes—
21#4984—21no—
22#4984C2212yes—
23#4976—21no—
24#4975C2211yescharacteristic
25Dic3×C4⋊C4—21no—
26Dic3×C4⋊C4C2212yes—
27Dic3×C4⋊C4C2211yescharacteristic
28Dic3×C4⋊C4C2211yescharacteristic
29C2×#807C2212yes—
30#4935C2211yescharacteristic
31C42×Dic3C2211yescharacteristic
32#4928C2211yescharacteristic
33#4649—21no—
34#4646—21no—
35C3×#340C2212yes—
36C3×#336C2211yescharacteristic
37#4551C2212yes—
38#4551C2212yes—
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C4⋊Q8—31no—
45C4×C4⋊Q8—31no—
46#2099—31no—
47#2099—31no—
48#2057—31no—
49#2057D311yescharacteristic
50#1988—32no—
51#1947—32no—
52#1918—32no—
53#1918—32no—
54#1831—31no—
55C22×C3×Q8—21no—
56C22×Dic6—21no—
57C22×C3×C8C2312yes—
58C3×C4⋊Q8—24no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C42C2311yescharacteristic
65Q8×Dic3—44no—
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×Dic3⋊C4—21no—
75C2×Dic3⋊C4—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C2×C4×Dic3C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86#888—22no—
87#888Q814yescomplemented
88#887—44no—
89C3×#107—24no—
90C3×#95—21no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96Dic3⋊C8—24no—
97#807—24no—
98C2×C4⋊Q8—31no—
99C2×C4⋊Q8C2×D311yescharacteristic
100C2×C4×Q8—61no—
101C2×C4×Q8—61no—
102C2×#107—32no—
103C2×#107C2×D312yes—
104C2×C4⋊C8—32no—
105#391—61no—
106#391—61no—
107#389—32no—
108#389—31no—
109#389—61no—
110#389—61no—
111#389C2×D311yescharacteristic
112#384—31no—
113#384—31no—
114C4×C4⋊C4—31no—
115C4×C4⋊C4—31no—
116C4×C4⋊C4—32no—
117C4×C4⋊C4—61no—
118C4×C4⋊C4—61no—
119C43—31no—
120#340—32no—
121#340—32no—
122#340—32no—
123#340—32no—
124#340—32no—
125#340C2×D312yes—
126#336—32no—
127#336—31no—
128#336C2×D311yescharacteristic
129C2×C3×Q8—24no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—22no—
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D411yescharacteristic
143C22×Dic3C2×D411yescharacteristic
144C22×Dic3C2×D411yescharacteristic
145C2×Dic6—24no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4C2×Q812yes—
157C3×C4⋊C4#4114yes—
158C3×C42—22no—
159C4⋊Dic3—21no—
160C4⋊Dic3—22no—
161C4⋊Dic3—24no—
162C4⋊Dic3—42no—
163C4⋊Dic3—42no—
164C4⋊Dic3—44no—
165C4⋊Dic3C2×Q812yes—
166Dic3⋊C4—24no—
167Dic3⋊C4—42no—
168Dic3⋊C4—42no—
169Dic3⋊C4—42no—
170C4×Dic3—22no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C4×Dic3—41no—
174C4×Dic3—42no—
175C4×Dic3—42no—
176C4×Dic3—44no—
177C4×Dic3D814yes—
178C4×Dic3C2×Q812yes—
179C4×Dic3#4112yes—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C22×Q8—21no—
184C22×Q8—61no—
185C22×C8—32no—
186C22×C8C22×D312yes—
187C4⋊Q8—24no—
188C4⋊Q8—34no—
189C4⋊Q8—62no—
190C4×Q8—124no—
191C4×Q8—124no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4C22×D311yescharacteristic
205C2×C4⋊C4C22×D312yes—
206C2×C4⋊C4C22×D311yescharacteristic
207C2×C42—31no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42—31no—
211C2×C42—31no—
212C2×C42—31no—
213C2×C42—61no—
214C2×C42—62no—
215C2×C42—61no—
216C2×C42C22×D311yescharacteristic
217#107—24no—
218#107—64no—
219C4⋊C8—64no—
220#95—21no—
221#95—61no—
222#95—62no—
223#95—61no—
224#95—62no—
225#95—61no—
226C23×C3C22×D411yescharacteristic
227C3×Q8—48no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4C2×D812yes—
242C2×C3×C4#13712yes—
243C2×C3×C4C22×Q811yescharacteristic
244C2×C3×C4C2×#4112yes—
245C2×C3×C4C2×#4111yescharacteristic
246C2×C3×C42−1+412yes—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—22no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3C22⋊Q812yes—
264C2×Dic3C22⋊Q812yes—
265C2×Dic3C2×D812yes—
266C2×Dic3C2×D812yes—
267C2×Dic3#13712yes—
268Dic6—48no—
269C3×C8—42no—
270C3⋊C8—42no—
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—64no—
275C2×Q8—122no—
276C2×Q8—122no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4C23×D311yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—62no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—32no—
301C4⋊C4—42no—
302C4⋊C4—64no—
303C4⋊C4—61no—
304C4⋊C4—62no—
305C4⋊C4—64no—
306C4⋊C4—122no—
307C4⋊C4—124no—
308C4⋊C4—122no—
309C4⋊C4—122no—
310C4⋊C4—122no—
311C4⋊C4—122no—
312C4⋊C4#23714yes—
313C4⋊C4D3×Q812yes—
314C42—22no—
315C42—34no—
316C42—32no—
317C42—32no—
318C42—62no—
319C42—62no—
320C42—62no—
321C42—121no—
322C42—124no—
323C42—122no—
324C42—122no—
325C22×C3C2×C22⋊Q811yescharacteristic
326C22×C3#52511yescharacteristic
327C22×C3#53611yescharacteristic
328C22×C3C22×D811yescharacteristic
329C22×C3C2×#13711yescharacteristic
330C22×C3C2×#13711yescharacteristic
331C22×C3#57511yescharacteristic
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—44no—
335C3×C4—42no—
336C3×C4—81no—
337C3×C4#55612yes—
338C3×C4#55712yes—
339Dic3—22no—
340Dic3—22no—
341Dic3—41no—
342Dic3—42no—
343Dic3—81no—
344Dic3—82no—
345Dic3#47414yes—
346C23C2×D3×D411yescharacteristic
347Q8—48no—
348Q8—128no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—62no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4D3×D812yes—
384C2×C4#91812yes—
385C2×C4C2×#23712yes—
386C2×C4C2×D3×Q811yescharacteristic
387C2×C4#100712yes—
388C2×C4D3×#4111yescharacteristic
389C8—42no—
390C8—122no—
391C2×C3#258111yescharacteristic
392C2×C3C2×#47411yescharacteristic
393C2×C3#307811yescharacteristic
394C2×C3#310011yescharacteristic
395C2×C3#311311yescharacteristic
396C2×C3#322611yescharacteristic
397C2×C3#323711yescharacteristic
398C22#558611yescharacteristic
399C22D3×C22⋊Q811yescharacteristic
400C22#572411yescharacteristic
401C22C2×D3×D811yescharacteristic
402C22C2×#91811yescharacteristic
403C22#577311yescharacteristic
404C22D3×#13711yescharacteristic
405C4—22no—
406C4—22no—
407C4—34no—
408C4—42no—
409C4—44no—
410C4—62no—
411C4—62no—
412C4—81no—
413C4—121no—
414C4—122no—
415C4—242no—
416C4—241no—
417C4#564112yes—
418C4#572112yes—
419C3#2927211yescharacteristic, complemented, Sylow
420C2#7719111yescharacteristic
421C2#7998811yescharacteristic
422C2#8017411yescharacteristic
423C2D3×#47411yescharacteristic
424C2#8110811yescharacteristic
425C2#8194111yescharacteristic
426C2#8198111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socle, prime radicalC23×C3C22×D4
hypercenter#2057D3
hypocenterC3#29272
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
87. #888Q8semidirect
419. C3#29272semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊Q82
#759672
C3×#20572
#717822
#706152
#706092
#696112
#688152
#688002
#686422
#686052
#292723