Finite Group Databasebeta

← #10877270

Subgroups — #10877270

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75965C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic
3C3×#2099C211yesmaximal, characteristic, complemented
4#69612C211yesmaximal, characteristic, complemented
5#69591C211yesmaximal, characteristic, complemented
6#68905C212yesmaximal, complemented
7#68879C212yesmaximal
8#68877C212yesmaximal
9#68798C212yesmaximal, complemented
10#68691C211yesmaximal, characteristic, complemented
11#68662C211yesmaximal, characteristic
12#27685—31nomaximal, Sylow
13C2×#897C2211yescharacteristic
14C2×#892C2212yes—
15C2×#887C2211yescharacteristic
16C2×C42×D3C2211yescharacteristic
17C3×#389C2211yescharacteristic
18C3×#384C2211yescharacteristic
19C2×(C2×D3)⋊C8C2212yes—
20C2×#818C2212yes—
21C2×#816C2212yes—
22C2×Dic3⋊C8C2212yes—
23#4989—21no—
24#4987—21no—
25#4987—21no—
26#4987C2211yescharacteristic
27#4977C2212yes—
28Dic3×C4⋊C4—21no—
29Dic3×C4⋊C4—21no—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4C2211yescharacteristic
32C2×#810C2212yes—
33C2×#808C2212yes—
34#4935C2211yescharacteristic
35#4929C2211yescharacteristic
36#4666—21no—
37#4654—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C22⋊Q8—31no—
45#2285—31no—
46#2100—31no—
47#2099D311yescharacteristic, complemented
48#2096—31no—
49#2095—31no—
50#1949—32no—
51#1873—32no—
52#1854—32no—
53#1805—32no—
54#1797—31no—
55C22×C4×D3D411yescharacteristic
56C22×C4×D3D411yescharacteristic
57C22×C4×D3C2311yescharacteristic
58C22×Dic6C2311yescharacteristic
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×Dic3⋊C4—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3C2311yescharacteristic
82C22×C3⋊C8C2312yes—
83#897—22no—
84#897Q814yes—
85#892—42no—
86#892—42no—
87#887—24no—
88C42×D3—24no—
89C3×#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96(C2×D3)⋊C8—24no—
97#818—42no—
98#816—42no—
99Dic3⋊C8—24no—
100#810—42no—
101#808—42no—
102C2×C22⋊Q8—31no—
103C2×C4×Q8—31no—
104C2×#117—32no—
105C22×C42—31no—
106C2×#107—32no—
107C2×#106—32no—
108C2×C4⋊C8—32no—
109C2×#103—32no—
110C2×#103—32no—
111C2×C22⋊C8—32no—
112#391—31no—
113#389C2×D311yescharacteristic
114#387—61no—
115#385—61no—
116#384—31no—
117#384C2×D311yescharacteristic
118#382—32no—
119#380—31no—
120#380—61no—
121#380—61no—
122C4×C4⋊C4—31no—
123C4×C4⋊C4—61no—
124C4×C4⋊C4—61no—
125C4×C4⋊C4—61no—
126C4×C22⋊C4—61no—
127#340—32no—
128#340—32no—
129#340C2×D312yes—
130#340C2×D312yes—
131#336—31no—
132#336C2×D311yescharacteristic
133C23×D3C2×D411yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3C2×D411yescharacteristic
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3C2×Q812yes—
155C2×C4×D3#4112yes—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—21no—
159C2×Dic6C2×Q812yes—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4C2×Q812yes—
171C3×C42—22no—
172(C2×D3)⋊C4—24no—
173(C2×D3)⋊C4—44no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—24no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—42no—
180C4⋊Dic3—42no—
181Dic3⋊C4—42no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C4×Dic3—44no—
185C4×Dic3—42no—
186C4×Dic3—42no—
187C4×Dic3#4114yes—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C22×Q8—31no—
192C23×C4—31no—
193C23×C4—31no—
194C23×C4—31no—
195C22×C8—32no—
196C22×C8C22×D312yes—
197C22⋊Q8—34no—
198C22⋊Q8—62no—
199C4×Q8—64no—
200#117—122no—
201#117—122no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4C22×D312yes—
212C2×C4⋊C4C22×D311yescharacteristic
213C2×C4⋊C4C22×D311yescharacteristic
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—61no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—64no—
221C2×C42—62no—
222C2×C42—61no—
223C2×C42—61no—
224C2×C42C22×D311yescharacteristic
225#107—122no—
226#106—122no—
227C4⋊C8—64no—
228#103—122no—
229#103—122no—
230C22⋊C8—64no—
231#95—21no—
232#95—61no—
233#95—61no—
234#95—62no—
235#95—62no—
236#95—61no—
237#95—61no—
238C23×C3C22×D411yescharacteristic
239C22×D3—21no—
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—21no—
245C22×D3C22⋊Q812yes—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4#13512yes—
260C2×C3×C4#13712yes—
261C2×C3×C4C22×Q811yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C4H(5,𝔽2)12yes—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—22no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3C22⋊Q812yes—
281C2×Dic3#12312yes—
282C2×Dic3#12312yes—
283C4×D3—24no—
284C4×D3—24no—
285C4×D3—44no—
286C4×D3—44no—
287Dic6—24no—
288Dic6—42no—
289C3×C8—42no—
290C3⋊C8—42no—
291C24—31no—
292C2×Q8—32no—
293C2×Q8—61no—
294C2×Q8—62no—
295C2×Q8—62no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—62no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4C23×D311yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—62no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—64no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—122no—
336C4⋊C4—122no—
337C4⋊C4—122no—
338C4⋊C4D3×Q812yes—
339C22⋊C4—64no—
340C22⋊C4—124no—
341C22⋊C4—122no—
342C42—22no—
343C42—34no—
344C42—62no—
345C42—62no—
346C42—122no—
347C42—124no—
348C42—122no—
349C22×C3C2×C22⋊Q811yescharacteristic
350C22×C3C2×#12311yescharacteristic
351C22×C3#53511yescharacteristic
352C22×C3C2×#13511yescharacteristic
353C22×C3C2×#13711yescharacteristic
354C22×C3#57511yescharacteristic
355C22×C3#57511yescharacteristic
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—81no—
367C3×C4—82no—
368C3×C4#55312yes—
369C3×C4#55412yes—
370Dic3—22no—
371Dic3—22no—
372Dic3—22no—
373Dic3—42no—
374Dic3—82no—
375Dic3—81no—
376C23—32no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23C2×D3×D411yescharacteristic
384Q8—64no—
385Q8—122no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—62no—
411C2×C4—124no—
412C2×C4—124no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4#91212yes—
422C2×C4#91512yes—
423C2×C4C2×D3×Q811yescharacteristic
424C2×C4D3×#4112yes—
425C2×C4D3×#4111yescharacteristic
426C2×C4#100912yes—
427C8—42no—
428C8—122no—
429C2×C3#248711yescharacteristic
430C2×C3#307611yescharacteristic
431C2×C3#307811yescharacteristic
432C2×C3#309111yescharacteristic
433C2×C3#309511yescharacteristic
434C2×C3#321011yescharacteristic
435C2×C3#321911yescharacteristic
436D3—42no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22D3×C22⋊Q811yescharacteristic
445C22D3×#12311yescharacteristic
446C22#567411yescharacteristic
447C22C2×#91211yescharacteristic
448C22C2×#91511yescharacteristic
449C22#577711yescharacteristic
450C22#577711yescharacteristic
451C4—22no—
452C4—22no—
453C4—42no—
454C4—62no—
455C4—62no—
456C4—62no—
457C4—81no—
458C4—82no—
459C4—122no—
460C4—242no—
461C4—241no—
462C4#569012yes—
463C4#570912yes—
464C3#2768511yescharacteristic, complemented, Sylow
465C2—122no—
466C2#7814611yescharacteristic
467C2#8096411yescharacteristic
468C2#8101111yescharacteristic
469C2#8128111yescharacteristic
470C2#8142811yescharacteristic
471C2#8156911yescharacteristic
472C2#8174211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2099C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2099D3
hypocenterC3#27685
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
3. C3×#2099C2semidirect
4. #69612C2semidirect
5. #69591C2semidirect
6. #68905C2semidirect
9. #68798C2semidirect
10. #68691C2semidirect
47. #2099D3semidirect
464. C3#27685semidirect

Maximal subgroups

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

subgroupindex
#759652
#743182
C3×#20992
#696122
#695912
#689052
#688792
#688772
#687982
#686912
#686622
#276853