Finite Group Databasebeta

← #10876158

Subgroups — #10876158

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#391C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic
3C3×#2095C211yesmaximal, characteristic, complemented
4#70632C212yesmaximal
5#70624C212yesmaximal, complemented
6#70266C212yesmaximal
7#70242C212yesmaximal, complemented
8#69670C211yesmaximal, characteristic, complemented
9#69590C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic
11#68636C211yesmaximal, characteristic, complemented
12#27586—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×#892—21no—
15C2×D3×C4⋊C4C2211yescharacteristic
16C2×C42×D3C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5229C2211yescharacteristic
22C2×#835C2212yes—
23C2×(C2×D3)⋊C8C2212yes—
24C2×#816C2212yes—
25#4987—21no—
26#4987C2212yes—
27#4977—21no—
28Dic3×C4⋊C4—21no—
29Dic3×C4⋊C4C2212yes—
30#4973C2211yescharacteristic
31C2×#810C2212yes—
32C2×#807C2212yes—
33#4935C2211yescharacteristic
34#4929C2211yescharacteristic
35D3×#95—21no—
36D3×#95—21no—
37#4646—21no—
38#4646—21no—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C2×#391—31no—
45#2285—31no—
46#2095—31no—
47#2095D311yescharacteristic, complemented
48#2072—31no—
49#2072—31no—
50#1928—32no—
51#1928—32no—
52#1852—32no—
53#1817—31no—
54#1787—32no—
55C22×C3×Q8C2311yescharacteristic
56C2×D3×Q8—22no—
57C2×D3×Q8—22no—
58C2×D3×Q8—24no—
59C2×D3×Q8—42no—
60C22×C4×D3—21no—
61C22×C4×D3—21no—
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C22×Dic6—21no—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×Dic3⋊C4—21no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3C2311yescharacteristic
83C22×C3⋊C8C2312yes—
84#892—44no—
85D3×C4⋊C4—42no—
86C42×D3—24no—
87C3×#107—42no—
88C3×#103—42no—
89C3×#95—21no—
90C3×#95—21no—
91#835—24no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100(C2×D3)⋊C8—24no—
101#816—24no—
102#810—42no—
103#807—42no—
104C23×Q8—31no—
105C2×#117—61no—
106C22×C4⋊C4—31no—
107C22×C42—31no—
108C2×#107—32no—
109C2×#107C2×D312yes—
110C2×#103—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×#103C2×D312yes—
114C2×C22⋊C8—32no—
115#391—31no—
116#391—31no—
117#391—31no—
118#391—61no—
119#391—61no—
120#391C2×D311yescharacteristic
121#384—31no—
122#384C2×D311yescharacteristic
123#382—61no—
124#380—32no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—61no—
128C2×#95—61no—
129C2×#95—61no—
130#340—32no—
131#340—32no—
132#336—32no—
133#336—31no—
134#336C2×D311yescharacteristic
135C23×D3C2×D411yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8#4112yes—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—22no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152D3×Q8—48no—
153D3×Q8—48no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—22no—
159C2×C4×D3—24no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—41no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3Q1614yes—
169C2×C4×D3#4114yes—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—24no—
173C2×Dic6—21no—
174C2×Dic6—42no—
175C2×Dic6—42no—
176C2×Dic6#4112yes—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C42—22no—
186(C2×D3)⋊C4—44no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192Dic3⋊C4—42no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—44no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C22×Q8—31no—
201C22×Q8—61no—
202C22×Q8—64no—
203C22×Q8—62no—
204C22×Q8—62no—
205C22×Q8—122no—
206C22×Q8C22×D311yescharacteristic
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—31no—
210C23×C4—61no—
211C23×C4—61no—
212C22×C8—32no—
213C22×C8C22×D312yes—
214#117—124no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—122no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—62no—
226C2×C42—31no—
227C2×C42—31no—
228C2×C42—31no—
229C2×C42—64no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#107—42no—
233#107—122no—
234#103—42no—
235#103—64no—
236#103—64no—
237#103—122no—
238C22⋊C8—64no—
239#95—21no—
240#95—21no—
241#95—61no—
242#95—61no—
243#95—61no—
244#95—61no—
245#95—62no—
246#95—62no—
247#95—61no—
248#95—61no—
249C23×C3C22×D411yescharacteristic
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—21no—
254C22×D3#12312yes—
255C22×D3C2×Q1612yes—
256C22×D3AGL(1,ℤ/8)12yes—
257C3×Q8—24no—
258C3×Q8—42no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4C2×Q1612yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4112yes—
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4H(5,𝔽2)12yes—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—22no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3#12312yes—
295C2×Dic3C2×Q1612yes—
296C4×D3—28no—
297C4×D3—24no—
298C4×D3—24no—
299C4×D3—44no—
300C4×D3—48no—
301C4×D3—84no—
302Dic6—24no—
303Dic6—48no—
304Dic6—42no—
305C3×C8—42no—
306C3⋊C8—42no—
307C24—31no—
308C2×Q8—21no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—32no—
312C2×Q8—64no—
313C2×Q8—61no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8—122no—
317C2×Q8—122no—
318C2×Q8—128no—
319C2×Q8—128no—
320C2×Q8#23912yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—34no—
327C22×C4—34no—
328C22×C4—31no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—62no—
338C22×C4—61no—
339C22×C4—62no—
340C22×C4—121no—
341C22×C4—122no—
342C22×C4—122no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4C23×D311yescharacteristic
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—62no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—122no—
366C22⋊C4—124no—
367C42—22no—
368C42—62no—
369C42—62no—
370C42—62no—
371C42—124no—
372C22×C3C2×#12311yescharacteristic
373C22×C3#52511yescharacteristic
374C22×C3#53511yescharacteristic
375C22×C3C22×Q1611yescharacteristic
376C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
377C22×C3C2×#13711yescharacteristic
378C22×C3#57511yescharacteristic
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—22no—
384C2×D3—22no—
385C2×D3#48414yes—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—81no—
390C3×C4—82no—
391C3×C4#55112yes—
392C3×C4#55112yes—
393Dic3—22no—
394Dic3—22no—
395Dic3—41no—
396Dic3—42no—
397Dic3—82no—
398Dic3—81no—
399C23—32no—
400C23—32no—
401C23—32no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23C2×D3×D411yescharacteristic
407Q8—24no—
408Q8—42no—
409Q8—64no—
410Q8—128no—
411Q8—122no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—68no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—62no—
438C2×C4—124no—
439C2×C4—128no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4—244no—
449C2×C4D3×Q1612yes—
450C2×C4#91812yes—
451C2×C4#100412yes—
452C2×C4C2×#23911yescharacteristic
453C2×C4D3×#4112yes—
454C2×C4D3×#4111yescharacteristic
455C8—42no—
456C8—122no—
457C2×C3#248211yescharacteristic
458C2×C3C2×#48411yescharacteristic
459C2×C3#309511yescharacteristic
460C2×C3#310411yescharacteristic
461C2×C3#311211yescharacteristic
462C2×C3#320911yescharacteristic
463C2×C3#321311yescharacteristic
464D3—24no—
465C22—34no—
466C22—62no—
467C22—62no—
468C22—62no—
469C22—62no—
470C22—62no—
471C22—62no—
472C22#562711yescharacteristic
473C22D3×#12311yescharacteristic
474C22#571511yescharacteristic
475C22C2×D3×Q1611yescharacteristic
476C22C2×#91811yescharacteristic
477C22D3×AGL(1,ℤ/8)11yescharacteristic
478C22#577311yescharacteristic
479C4—22no—
480C4—22no—
481C4—42no—
482C4—62no—
483C4—62no—
484C4—81no—
485C4—82no—
486C4—122no—
487C4—121no—
488C4—241no—
489C4—242no—
490C4#563712yes—
491C4#565912yes—
492C3#2758611yescharacteristic, complemented, Sylow
493C2—64no—
494C2#7810511yescharacteristic
495C2#8109211yescharacteristic
496C2#8109411yescharacteristic
497C2D3×#48411yescharacteristic
498C2#8146911yescharacteristic
499C2#8180211yescharacteristic
500C2#8184911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2095C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2095D3
hypocenterC3#27586
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
3. C3×#2095C2semidirect
5. #70624C2semidirect
7. #70242C2semidirect
8. #69670C2semidirect
9. #69590C2semidirect
11. #68636C2semidirect
47. #2095D3semidirect
492. C3#27586semidirect

Maximal subgroups

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

subgroupindex
D3×#3912
#743182
C3×#20952
#706322
#706242
#702662
#702422
#696702
#695902
#686642
#686362
#275863