Finite Group Databasebeta

← #10824702

Subgroups — #10824702

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic12C211yesmaximal, characteristic, complemented
2C2×#4690C211yesmaximal, characteristic
3#73762C212yesmaximal
4C3×C4×#107C211yesmaximal, characteristic, complemented
5#70643C212yesmaximal, complemented
6#70635C212yesmaximal
7#69475C212yesmaximal, complemented
8#68888C211yesmaximal, characteristic
9#68864C211yesmaximal, characteristic, complemented
10#68838C211yesmaximal, characteristic, complemented
11#68830C211yesmaximal, characteristic
12#23705—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C2×#897C2212yes—
15C2×#872C2211yescharacteristic
16C2×C4×Dic6C2212yes—
17C2×C3×#107C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2212yes—
20C2×(C2×D3)⋊C8C2211yescharacteristic
21C2×(C2×D3)⋊C8C2211yescharacteristic
22C2×#818C2211yescharacteristic
23C2×#816C2212yes—
24C2×Dic3⋊C8C2211yescharacteristic
25C2×Dic3⋊C8C2211yescharacteristic
26#4987C2212yes—
27#4984C2212yes—
28#4973C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31#4940—22no—
32#4928—22no—
33C2×#803C2211yescharacteristic
34C4×Dic12—24no—
35#4690—24no—
36#4670—22no—
37#4653—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40#4529C2212yes—
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#433—31no—
44#2323—32no—
45#2086—31no—
46#2075—31no—
47#2068—32no—
48#2056—31no—
49#2036—32no—
50#2001—31no—
51C4×#107D311yescharacteristic, complemented
52#1763—32no—
53C22×C4×D3C2311yescharacteristic
54C22×Dic6C2312yes—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C42—22no—
60C2×C3×C42C2311yescharacteristic
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4D411yescharacteristic
64C2×(C2×D3)⋊C4D411yescharacteristic
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4D411yescharacteristic
70C2×Dic3⋊C4D411yescharacteristic
71C2×C4×Dic3—22no—
72C2×C4×Dic3C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C2×Dic12—24no—
76C2×Dic12—22no—
77#897—24no—
78#897—24no—
79#872—22no—
80#872—22no—
81C4×Dic6—24no—
82C4×Dic6—24no—
83C3×#107—24no—
84C3×C4×C8D414yes—
85C3×#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#818—22no—
93#816—24no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96#810—24no—
97#810—24no—
98#803—22no—
99C22×Q16—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C2×#117—31no—
103C4×Q16—64no—
104#433—64no—
105C2×#107—31no—
106C2×#107C2×D312yes—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×#97—31no—
115C2×C4×C8C2×D311yescharacteristic
116#391—32no—
117#389—32no—
118#389—62no—
119#388—62no—
120#388—62no—
121#387—62no—
122#380—32no—
123C4×C4⋊C4C2×D312yes—
124#340—32no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D311yescharacteristic
128C23×D3C2×D411yescharacteristic
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3C2×D411yescharacteristic
137C22×Dic3C2×D411yescharacteristic
138C22×Dic3C2×D411yescharacteristic
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—22no—
142C2×C4×D3—22no—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6#4114yes—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8C2×D412yes—
152C2×C3×C8C2×D412yes—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4#4114yes—
157C3×C42—82no—
158C3×C42C2×D412yes—
159C3×C42#4112yes—
160(C2×D3)⋊C4—28no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—28no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—82no—
168Dic3⋊C4—22no—
169Dic3⋊C4—22no—
170Dic3⋊C4—44no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C4×Dic3—44no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178Dic12—28no—
179C22×Q8—32no—
180C23×C4—31no—
181C2×Q16—64no—
182C2×Q16—62no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187C22⋊Q8—64no—
188C22⋊Q8—64no—
189C4×Q8—64no—
190C4×Q8—64no—
191#117—62no—
192#117—62no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4C22×D312yes—
200C2×C4⋊C4C22×D312yes—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—31no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C42—22no—
206C2×C42—31no—
207C2×C42—62no—
208C2×C42C22×D311yescharacteristic
209#107—24no—
210#107—62no—
211C4⋊C8—62no—
212C4⋊C8—62no—
213#103—64no—
214#103—64no—
215#103—64no—
216C22⋊C8—62no—
217C22⋊C8—62no—
218#97—62no—
219C4×C8D1214yes—
220#95—22no—
221#95—62no—
222#95—62no—
223#95—62no—
224#95—62no—
225C23×C3C22×D411yescharacteristic
226C22×D3—21no—
227C22×D3—22no—
228C22×D3—21no—
229C22×D3—21no—
230C22×D3—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12112yes—
241C2×C3×C4#12112yes—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4112yes—
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—22no—
250C2×Dic3—24no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C4×D3—41no—
260C4×D3—42no—
261C4×D3—41no—
262Dic6—24no—
263Dic6—24no—
264Dic6—42no—
265Dic6—42no—
266C3×C8—22no—
267C3×C8—22no—
268C3⋊C8—41no—
269C3⋊C8—41no—
270C24—31no—
271C2×Q8—34no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×Q8—62no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—62no—
283C22×C4—61no—
284C22×C4—62no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4D3×D411yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4C23×D311yescharacteristic
291Q16—68no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8C2×D1212yes—
299C2×C8C2×D1212yes—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—68no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—124no—
309C4⋊C4—242no—
310C4⋊C4#23514yes—
311C22⋊C4—68no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—242no—
315C42—61no—
316C42—61no—
317C42—82no—
318C42—124no—
319C42C2×D1212yes—
320C42#23912yes—
321C22×C3C2×#12111yescharacteristic
322C22×C3#52512yes—
323C22×C3#57611yescharacteristic
324C22×C3#57811yescharacteristic
325C22×C3#57812yes—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—82no—
335C3×C4#52511yescharacteristic
336C3×C4#52912yes—
337C3×C4#53211yescharacteristic
338Dic3—44no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—82no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—62no—
347C23—62no—
348C23C2×D3×D411yescharacteristic
349Q8—64no—
350Q8—64no—
351Q8—122no—
352Q8—122no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—64no—
369C2×C4—62no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4#89512yes—
379C2×C4#89512yes—
380C2×C4C22×D1211yescharacteristic
381C2×C4C2×#23512yes—
382C2×C4C2×#23911yescharacteristic
383C2×C4D3×#4112yes—
384C8—22no—
385C8—22no—
386C8—121no—
387C8—121no—
388C2×C3#239211yescharacteristic
389C2×C3#305911yescharacteristic
390C2×C3#306111yescharacteristic
391C2×C3#312412yes—
392C2×C3#312712yes—
393D3—81no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—122no—
398C22—122no—
399C22C2×#89511yescharacteristic
400C22#554712yes—
401C22#575711yescharacteristic
402C22#576611yescharacteristic
403C22#577112yes—
404C4—22no—
405C4—22no—
406C4—44no—
407C4—82no—
408C4—121no—
409C4—121no—
410C4—124no—
411C4—121no—
412C4—242no—
413C4#547711yescharacteristic
414C4#552411yescharacteristic
415C4#558012yes—
416C3#2370511yescharacteristic, complemented, Sylow
417C2—241no—
418C2#7377611yescharacteristic
419C2#7915011yescharacteristic
420C2#7918411yescharacteristic
421C2#7959712yes—
422C2#7972412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#107C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC4×#107D3
hypocenterC3#23705
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic12C2semidirect
4. C3×C4×#107C2semidirect
5. #70643C2semidirect
7. #69475C2semidirect
9. #68864C2semidirect
10. #68838C2semidirect
51. C4×#107D3semidirect
416. C3#23705semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic122
C2×#46902
#737622
C3×C4×#1072
#706432
#706352
#694752
#688882
#688642
#688382
#688302
#237053