Finite Group Databasebeta

← #10801881

Subgroups — #10801881

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4867C211yesmaximal, characteristic
2C2×C4×Dic12C211yesmaximal, characteristic, complemented
3C2×#4689C211yesmaximal, characteristic
4#73762C212yesmaximal
5C3×C4×#106C211yesmaximal, characteristic, complemented
6#70578C211yesmaximal, characteristic
7#70575C211yesmaximal, characteristic, complemented
8#70562C211yesmaximal, characteristic, complemented
9#69475C212yesmaximal, complemented
10#68882C212yesmaximal
11#68851C212yesmaximal, complemented
12#21537—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×#897C2212yes—
17C2×#872C2211yescharacteristic
18C2×C4×Dic6C2212yes—
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×C4×C4⋊C4C2212yes—
23C2×#818C2211yescharacteristic
24C2×#816C2212yes—
25C2×#815C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27#4987C2212yes—
28#4984C2212yes—
29#4973C2212yes—
30C2×#810C2212yes—
31C2×#810C2212yes—
32#4940—22no—
33#4928—22no—
34C2×#803C2211yescharacteristic
35#4867C418yescomplemented
36C4×Dic12—24no—
37#4689—24no—
38#4670—22no—
39#4653—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#470—31no—
44C2×C4×Q16—31no—
45C2×#405—31no—
46#2323—32no—
47#1971—31no—
48#1944—31no—
49#1933—32no—
50#1892—32no—
51#1809—31no—
52C4×#106D311yescharacteristic, complemented
53#1763—32no—
54C22×C4×D3C2311yescharacteristic
55C22×Dic6C2312yes—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C42—22no—
61C2×C3×C42C2311yescharacteristic
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4D411yescharacteristic
65C2×(C2×D3)⋊C4D411yescharacteristic
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2312yes—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4D411yescharacteristic
71C2×Dic3⋊C4D411yescharacteristic
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C2×Dic12—22no—
77C2×Dic12—22no—
78C2×Dic12C2×C414yes—
79C2×ΓL(1,𝔽25)—22no—
80C2×ΓL(1,𝔽25)—22no—
81C2×ΓL(1,𝔽25)C2×C414yes—
82C2×D3×C8—22no—
83C2×D3×C8—22no—
84C2×D3×C8—22no—
85#897—24no—
86#897C2×C418yes—
87#872—22no—
88#872—22no—
89C4×Dic6—24no—
90C4×Dic6—24no—
91C3×#106—22no—
92C3×#106C2×C414yes—
93C3×C4×C8D414yes—
94C3×#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#818—22no—
100#816—24no—
101#815—22no—
102C8×Dic3—22no—
103#810—24no—
104#810C2×C418yes—
105#803—22no—
106C22×Q16—31no—
107C22×M16—31no—
108C23×C8—31no—
109C2×C22⋊Q8—32no—
110C2×C4×Q8—32no—
111C2×#117—31no—
112#470—38no—
113C4×Q16—64no—
114C2×#107—31no—
115C2×#106C2×D311yescharacteristic
116C2×#106C2×D311yescharacteristic
117C2×#103—32no—
118C2×#103—32no—
119C2×#103—32no—
120#405—64no—
121C2×#97—31no—
122C2×#97—31no—
123C2×C4×C8—31no—
124C2×C4×C8C2×D311yescharacteristic
125#391—32no—
126#389—32no—
127#389—62no—
128#388—62no—
129#388—62no—
130#387—62no—
131#380—32no—
132C4×C4⋊C4C2×D312yes—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136C23×D3C2×D411yescharacteristic
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C22×Dic3C2×D411yescharacteristic
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3C22×C412yes—
152C2×C4×D3#4112yes—
153C2×Dic6—22no—
154C2×Dic6—22no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6C22×C414yes—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8C22×C412yes—
162C2×C3×C8C2×D412yes—
163C2×C3×C8C2×D412yes—
164C2×C3×C8#4112yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4C22×C414yes—
169C3×C42—82no—
170C3×C42C2×D412yes—
171C3×C42#4112yes—
172(C2×D3)⋊C4—28no—
173(C2×D3)⋊C4—22no—
174(C2×D3)⋊C4—22no—
175(C2×D3)⋊C4—82no—
176C4⋊Dic3—28no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—82no—
180Dic3⋊C4—22no—
181Dic3⋊C4—22no—
182Dic3⋊C4—44no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—44no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8C22×C412yes—
191C2×C3⋊C8#4112yes—
192Dic12—28no—
193ΓL(1,𝔽25)—28no—
194D3×C8—28no—
195C22×Q8—32no—
196C23×C4—31no—
197C2×Q16—34no—
198C2×Q16—62no—
199C2×Q16—62no—
200C2×M16—34no—
201C2×M16—62no—
202C2×M16—62no—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8—62no—
206C22×C8—62no—
207C22×C8—62no—
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C22⋊Q8—38no—
211C22⋊Q8—64no—
212C4×Q8—64no—
213C4×Q8—64no—
214#117—62no—
215#117—62no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D312yes—
224C2×C22⋊C4—31no—
225C2×C22⋊C4—31no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C42—22no—
229C2×C42—31no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#107—62no—
233#106—22no—
234#106C4×D314yes—
235#103—38no—
236#103—64no—
237#103—64no—
238#97—62no—
239#97—62no—
240C4×C8—62no—
241C4×C8D1214yes—
242#95—22no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247C23×C3C22×D411yescharacteristic
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—22no—
251C22×D3—22no—
252C22×D3C4×D412yes—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4#12112yes—
262C2×C3×C4#12112yes—
263C2×C3×C4C23×C411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—24no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3C4×D412yes—
282C4×D3—22no—
283C4×D3—22no—
284C4×D3—22no—
285C4×D3—22no—
286Dic6—24no—
287Dic6—24no—
288Dic6—42no—
289Dic6—42no—
290C3×C8—22no—
291C3×C8C4×D414yes—
292C3⋊C8—22no—
293C3⋊C8—22no—
294C24—31no—
295C2×Q8—34no—
296C2×Q8—62no—
297C2×Q8—62no—
298C2×Q8—62no—
299C2×Q8—62no—
300C2×Q8—62no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4C23×D311yescharacteristic
317Q16—68no—
318M16—68no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—32no—
322C2×C8—32no—
323C2×C8—68no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8C2×C4×D312yes—
329C2×C8C2×D1212yes—
330C2×C8C2×D1212yes—
331C2×C8#23512yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—68no—
336C4⋊C4—62no—
337C4⋊C4—62no—
338C4⋊C4—62no—
339C4⋊C4—62no—
340C4⋊C4—124no—
341C4⋊C4—242no—
342C4⋊C4C2×C4×D314yes—
343C22⋊C4—68no—
344C22⋊C4—62no—
345C22⋊C4—62no—
346C22⋊C4—242no—
347C42—61no—
348C42—61no—
349C42—82no—
350C42—124no—
351C42C2×D1212yes—
352C42#23912yes—
353C22×C3C2×C4×D411yescharacteristic
354C22×C3C2×#12111yescharacteristic
355C22×C3#53211yescharacteristic
356C22×C3#57712yes—
357C22×C3#57812yes—
358C2×D3—24no—
359C2×D3—22no—
360C2×D3—42no—
361C2×D3—41no—
362C2×D3—41no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—44no—
366C3×C4—82no—
367C3×C4C2×C4×D411yescharacteristic
368C3×C4C4×#4111yescharacteristic
369C3×C4#52511yescharacteristic
370C3×C4#52911yescharacteristic
371Dic3—22no—
372Dic3—41no—
373Dic3—41no—
374Dic3—44no—
375Dic3—82no—
376C23—32no—
377C23—61no—
378C23—62no—
379C23—62no—
380C23—61no—
381C23C2×D3×D411yescharacteristic
382Q8—64no—
383Q8—64no—
384Q8—122no—
385Q8—122no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—32no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—64no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4#89512yes—
412C2×C4#89512yes—
413C2×C4C22×C4×D311yescharacteristic
414C2×C4C22×D1211yescharacteristic
415C2×C4C2×#23511yescharacteristic
416C2×C4C2×#23911yescharacteristic
417C2×C4D3×#4111yescharacteristic
418C2×C4D3×#4111yescharacteristic
419C8—22no—
420C8—62no—
421C8—62no—
422C8C4×D1214yes—
423C2×C3C4×#12111yescharacteristic
424C2×C3#294712yes—
425C2×C3#305911yescharacteristic
426C2×C3#306011yescharacteristic
427C2×C3#317812yes—
428D3—42no—
429C22—62no—
430C22—64no—
431C22—121no—
432C22—122no—
433C22—121no—
434C22C2×#89511yescharacteristic
435C22C4×D3×D411yescharacteristic
436C22#554811yescharacteristic
437C22#576212yes—
438C22#577112yes—
439C4—22no—
440C4—22no—
441C4—44no—
442C4—62no—
443C4—82no—
444C4—121no—
445C4—124no—
446C4—121no—
447C4—242no—
448C4C2×C4×D1211yescharacteristic
449C4#552411yescharacteristic
450C4C4×#23911yescharacteristic
451C4#558011yescharacteristic
452C3#2153711yescharacteristic, complemented, Sylow
453C2—122no—
454C2C4×#89511yescharacteristic
455C2#7912011yescharacteristic
456C2#7918411yescharacteristic
457C2#7964812yes—
458C2#7966812yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenterC4×#106D3
hypocenterC3#21537
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic12C2semidirect
5. C3×C4×#106C2semidirect
7. #70575C2semidirect
8. #70562C2semidirect
9. #69475C2semidirect
11. #68851C2semidirect
35. #4867C4semidirect
52. C4×#106D3semidirect
452. C3#21537semidirect

Maximal subgroups

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

subgroupindex
C2×#48672
C2×C4×Dic122
C2×#46892
#737622
C3×C4×#1062
#705782
#705752
#705622
#694752
#688822
#688512
#215373