Finite Group Databasebeta

← #10876189

Subgroups — #10876189

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4929C211yesmaximal, characteristic
2#82060C211yesmaximal, characteristic
3C3×#1817C211yesmaximal, characteristic
4#69963C211yesmaximal, characteristic
5#69963C211yesmaximal, characteristic
6#69928C211yesmaximal, characteristic
7#69927C211yesmaximal, characteristic
8#68693C212yesmaximal, complemented
9#68638C212yesmaximal, complemented
10#68628C212yesmaximal, complemented
11#68627C212yesmaximal, complemented
12#27589—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×C22⋊C8C2211yescharacteristic
18#5212C2211yescharacteristic
19#5212C2211yescharacteristic
20#5211—21no—
21C4×#217—21no—
22C2×C3⋊#95—21no—
23C2×C3⋊#95—21no—
24C2×#818C2212yes—
25C2×#817C2212yes—
26C2×#816C2212yes—
27C2×#816C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#4937—21no—
31#4936C2211yescharacteristic
32#4936C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34C4×C4⋊Dic3C2211yescharacteristic
35#4932—21no—
36#4929—21no—
37#4929—21no—
38#4929C2212yes—
39#4929C2212yes—
40C3×#336C2212yes—
41C3×#336C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C2×#391—31no—
45#2283—31no—
46#2097—32no—
47#2074—32no—
48#1928—32no—
49#1927—32no—
50#1853—31no—
51#1852—31no—
52#1817D311yescharacteristic
53#1787—31no—
54#1787—31no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6—24no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—42no—
65C22×Dic6C2312yes—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75C2×#217—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—42no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99#818—42no—
100#817—42no—
101#816—24no—
102#816—24no—
103#816—42no—
104#816—42no—
105C23×Q8—31no—
106C22×C4⋊C4—31no—
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×#103—32no—
114C2×C22⋊C8C2×D311yescharacteristic
115C2×C22⋊C8C2×D311yescharacteristic
116#391—32no—
117#391—32no—
118#391—61no—
119#391—61no—
120#383—61no—
121#382—31no—
122#382—31no—
123#381—61no—
124#380—31no—
125#380—31no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—31no—
128C4×C22⋊C4—61no—
129#376—61no—
130C2×#95—61no—
131C2×#95—61no—
132#340—32no—
133#340—32no—
134#336C2×D312yes—
135#336C2×D312yes—
136C24×C3C2×D411yescharacteristic
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—41no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4112yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C22×Dic3—24no—
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3—42no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—24no—
170C2×Dic6—22no—
171C2×Dic6—48no—
172C2×Dic6—48no—
173C2×Dic6—42no—
174C2×Dic6—42no—
175C2×Dic6#4114yes—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—21no—
188#217—42no—
189#217—42no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4×Dic3—42no—
198C4×Dic3—42no—
199C22×Q8—32no—
200C22×Q8—64no—
201C22×Q8—61no—
202C22×Q8—62no—
203C22×Q8—62no—
204C22×Q8—122no—
205C23×C4—61no—
206C23×C4—61no—
207C23×C4D1211yescharacteristic
208C23×C4D1211yescharacteristic
209C23×C4C22×D311yescharacteristic
210C22×C8C22×D312yes—
211C22×C8C22×D312yes—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—122no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—61no—
224C2×C42—61no—
225C2×C42D1211yescharacteristic
226C2×C42D1211yescharacteristic
227C2×C42C22×D311yescharacteristic
228C2×C42C22×D311yescharacteristic
229#107—122no—
230#106—122no—
231#103—64no—
232#103—64no—
233#103—122no—
234#103—122no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237#95—62no—
238#95—62no—
239#95—62no—
240#95—62no—
241#95—61no—
242#95—61no—
243#95—61no—
244#95—61no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—61no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3#12312yes—
254C23×C3#13512yes—
255C23×C3#13712yes—
256C23×C3C22×D411yescharacteristic
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4#12312yes—
275C2×C3×C4#13712yes—
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4112yes—
279C2×C3×C4H(5,𝔽2)12yes—
280C2×Dic3—24no—
281C2×Dic3—48no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—84no—
300Dic6—24no—
301Dic6—24no—
302Dic6—48no—
303Dic6—42no—
304Dic6—42no—
305C3×C8—42no—
306C3×C8—42no—
307C24C2×D1211yescharacteristic
308C2×Q8—34no—
309C2×Q8—64no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×Q8—128no—
316C2×Q8—128no—
317C2×Q8—122no—
318C2×Q8—122no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—64no—
328C22×C4—62no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4C2×D1211yescharacteristic
340C22×C4C2×D1211yescharacteristic
341C22×C4C2×D1211yescharacteristic
342C22×C4C2×D1211yescharacteristic
343C22×C4C2×D1211yescharacteristic
344C22×C4#23712yes—
345C22×C4#23712yes—
346C22×C4C23×D311yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C42—22no—
363C42—22no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—122no—
369C42—122no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3#48514yes—
377C22×C3C2×#12311yescharacteristic
378C22×C3#52511yescharacteristic
379C22×C3#53511yescharacteristic
380C22×C3C2×#13511yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C22×C3#57511yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4#55112yes—
390C3×C4#55112yes—
391Dic3—44no—
392Dic3—81no—
393Dic3—81no—
394Dic3—81no—
395Dic3—82no—
396Dic3—81no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23#88612yes—
402C23#90412yes—
403C23#90912yes—
404C23C22×D1211yescharacteristic
405Q8—64no—
406Q8—64no—
407Q8—128no—
408Q8—122no—
409Q8—122no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—64no—
428C2×C4—128no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—244no—
447C2×C4#88612yes—
448C2×C4#90912yes—
449C2×C4C2×#23711yescharacteristic
450C2×C4C2×#23711yescharacteristic
451C2×C4D3×#4112yes—
452C2×C4#100912yes—
453C8—42no—
454C8—42no—
455C2×C3—24no—
456C2×C3#248211yescharacteristic
457C2×C3C2×#48511yescharacteristic
458C2×C3#309511yescharacteristic
459C2×C3#310811yescharacteristic
460C2×C3#311711yescharacteristic
461C2×C3#321111yescharacteristic
462C2×C3#321811yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22#472314yes—
470C22C2×#88611yescharacteristic
471C22#552411yescharacteristic
472C22#552511yescharacteristic
473C22C2×#90411yescharacteristic
474C22C2×#90911yescharacteristic
475C22C2×#90911yescharacteristic
476C22#574811yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—124no—
483C4—242no—
484C4—241no—
485C4—241no—
486C4—241no—
487C4—241no—
488C4#565812yes—
489C4#565812yes—
490C3#2758911yescharacteristic, complemented, Sylow
491C2—24no—
492C2#7412911yescharacteristic
493C2C2×#472311yescharacteristic
494C2#7862511yescharacteristic
495C2#7930711yescharacteristic
496C2#7931111yescharacteristic
497C2#7935711yescharacteristic
498C2#7935811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1817D3
hypocenterC3#27589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #68693C2semidirect
9. #68638C2semidirect
10. #68628C2semidirect
11. #68627C2semidirect
490. C3#27589semidirect

Maximal subgroups

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

subgroupindex
C2×#49292
#820602
C3×#18172
#699632
#699282
#699272
#686932
#686382
#686282
#686272
#275893