Finite Group Databasebeta

← #10690190

Subgroups — #10690190

Order 768.

836 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74004C211yesmaximal, characteristic
2C2×#4673C211yesmaximal, characteristic
3#69963C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic
5#69525C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9016—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×#830C2211yescharacteristic
18C2×#816—22no—
19#4990—22no—
20C2×#810—22no—
21#4955—21no—
22Dic3×C22⋊C4—21no—
23#4673—22no—
24#4673C414yescomplemented
25#4670—22no—
26#4468—22no—
27C2×#393—31no—
28#2312—31no—
29#1787—31no—
30#1787—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C22×Dic6—21no—
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—44no—
45C22×Dic6C2×C412yes—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—44no—
58C2×(C2×D3)⋊C4C2×C412yes—
59C2×(C2×D3)⋊C4D412yes—
60C2×C4⋊Dic3D412yes—
61C2×C4×Dic3—22no—
62C22×C3⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95C2×C412yes—
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95—24no—
71#830—21no—
72#830—21no—
73#816—44no—
74#810—44no—
75C23×Q8—31no—
76C22×C22⋊C4—31no—
77C22×C22⋊C4—61no—
78C2×#103—62no—
79C2×#103—62no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#393—34no—
83#393—62no—
84#388—62no—
85#388—62no—
86#380—31no—
87#380C2×D311yescharacteristic
88#379—61no—
89C4×C22⋊C4—61no—
90C2×#95—61no—
91C2×#95C2×D311yescharacteristic
92#323—62no—
93C24×C3C2×D411yescharacteristic
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—42no—
101C23×D3—42no—
102C23×D3C22⋊C412yes—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—24no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C22×Dic3—42no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—42no—
125C22×Dic3C22⋊C412yes—
126C2×Dic6—24no—
127C2×Dic6—24no—
128C2×Dic6—22no—
129C2×Dic6—22no—
130C2×Dic6—44no—
131C2×Dic6—44no—
132C2×Dic6—44no—
133C2×Dic6—44no—
134C2×Dic6—42no—
135C2×Dic6—42no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×C3×C8—42no—
139C2×C3×C8—41no—
140C3×C4⋊C4—42no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—41no—
144#217—41no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—48no—
150(C2×D3)⋊C4—82no—
151(C2×D3)⋊C4—82no—
152C4⋊Dic3—42no—
153C4×Dic3—82no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C25—31no—
157C22×Q8—32no—
158C22×Q8—61no—
159C22×Q8—61no—
160C22×Q8—61no—
161C22×Q8—61no—
162C22×Q8—61no—
163C22×Q8—124no—
164C23×C4—21no—
165C23×C4—31no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C22×C8—21no—
169C22×C8—61no—
170C2×C4⋊C4—32no—
171C2×C4⋊C4D1212yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—32no—
174C2×C22⋊C4—32no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—64no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—62no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—124no—
184C2×C42—62no—
185#103—124no—
186#103—124no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191#95—21no—
192#95—21no—
193#95—64no—
194#95—62no—
195#95—61no—
196#95C4×D312yes—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3C2×C22⋊C411yescharacteristic
200C23×C3#12011yescharacteristic
201C23×C3#12311yescharacteristic
202C23×C3#13511yescharacteristic
203C23×C3#13511yescharacteristic
204C23×C3#13711yescharacteristic
205C23×C3#13711yescharacteristic
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—44no—
212C22×D3—44no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—81no—
220C22×D3—81no—
221C22×D3—82no—
222C22×D3—82no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4—82no—
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×Dic3—24no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—44no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—81no—
253Dic6—24no—
254Dic6—24no—
255Dic6—44no—
256Dic6—44no—
257Dic6—42no—
258Dic6—42no—
259C3×C8—81no—
260C3⋊C8—81no—
261C24—32no—
262C24—61no—
263C24—61no—
264C24—61no—
265C24—61no—
266C24—61no—
267C24—61no—
268C24—122no—
269C24—122no—
270C24D3×D411yescharacteristic
271C2×Q8—64no—
272C2×Q8—64no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—122no—
276C2×Q8—122no—
277C2×Q8—124no—
278C2×Q8—124no—
279C2×Q8—124no—
280C2×Q8—124no—
281C2×Q8—121no—
282C2×Q8—121no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—32no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—64no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—122no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—122no—
300C22×C4C2×C4×D311yescharacteristic
301C22×C4C2×D1211yescharacteristic
302C22×C4#23511yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4#23911yescharacteristic
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—122no—
309C2×C8—121no—
310C4⋊C4—42no—
311C4⋊C4—122no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—128no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—242no—
322C22⋊C4—242no—
323C42—242no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3#38611yescharacteristic
333C22×C3#41611yescharacteristic
334C22×C3#41911yescharacteristic
335C22×C3#45211yescharacteristic
336C22×C3#45211yescharacteristic
337C22×C3#45512yes—
338C22×C3B(4,𝔽2)12yes—
339C22×C3#48511yescharacteristic
340C22×C3#48511yescharacteristic
341C2×D3—44no—
342C2×D3—42no—
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—82no—
346C2×D3—82no—
347C2×D3—82no—
348C2×D3—82no—
349C2×D3—82no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—82no—
355C3×C4—82no—
356Dic3—42no—
357Dic3—44no—
358Dic3—82no—
359C23—22no—
360C23—22no—
361C23—62no—
362C23—62no—
363C23—62no—
364C23—122no—
365C23—122no—
366C23—124no—
367C23—124no—
368C23—121no—
369C23—121no—
370C23—122no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23—241no—
375C23—241no—
376C23—242no—
377C23—242no—
378C23D3×C22⋊C411yescharacteristic
379C23#88211yescharacteristic
380C23#89411yescharacteristic
381C23#91211yescharacteristic
382C23#91511yescharacteristic
383C23#91611yescharacteristic
384C23#91811yescharacteristic
385Q8—64no—
386Q8—64no—
387Q8—124no—
388Q8—124no—
389Q8—122no—
390Q8—122no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—64no—
404C2×C4—62no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—122no—
408C2×C4—124no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—241no—
417C2×C4C4×D1211yescharacteristic
418C2×C4#88411yescharacteristic
419C2×C4#89311yescharacteristic
420C2×C4#89511yescharacteristic
421C8—81no—
422C8—241no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3#160612yes—
428C2×C3#160612yes—
429C2×C3#187311yescharacteristic
430C2×C3#188511yescharacteristic
431C2×C3#188911yescharacteristic
432D3—82no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—124no—
442C22—122no—
443C22—122no—
444C22—122no—
445C22—242no—
446C22—242no—
447C22—242no—
448C22—242no—
449C22—242no—
450C22#466511yescharacteristic
451C22#474112yes—
452C22#477211yescharacteristic
453C22#478311yescharacteristic
454C22#480211yescharacteristic
455C22#480811yescharacteristic
456C22#482512yes—
457C22#486811yescharacteristic
458C22#488811yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—82no—
464C4—82no—
465C4—124no—
466C4—122no—
467C4—242no—
468C3#901611yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—242no—
474C2#6828712yes—
475C2#6831912yes—
476C2#6875911yescharacteristic
477C2#6887911yescharacteristic
478C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9016
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69963C2semidirect
5. #69525C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
24. #4673C4semidirect
33. #1294D3semidirect
468. C3#9016semidirect

Maximal subgroups

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

subgroupindex
#740042
C2×#46732
#699632
C2×#44682
#695252
C3×#12942
#676122
#90163