Finite Group Databasebeta

← #10876170

Subgroups — #10876170

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76478C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic, complemented
3C3×#2096C211yesmaximal, characteristic, complemented
4#70580C212yesmaximal, complemented
5#70569C212yesmaximal, complemented
6#70267C212yesmaximal
7#70242C212yesmaximal, complemented
8#69670C211yesmaximal, characteristic, complemented
9#69586C211yesmaximal, characteristic, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68635C211yesmaximal, characteristic, complemented
12#27587—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C2×#892—21no—
15C2×#892C2211yescharacteristic
16C2×C42×D3C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5229C2211yescharacteristic
22C2×#835C2212yes—
23C2×#821C2212yes—
24C2×(C2×D3)⋊C8C2212yes—
25#4988C2211yescharacteristic
26#4987—21no—
27#4987C2212yes—
28#4977—21no—
29Dic3×C4⋊C4—21no—
30Dic3×C4⋊C4C2212yes—
31C2×#809C2212yes—
32C2×#807C2212yes—
33#4939C2211yescharacteristic
34#4935C2211yescharacteristic
35#4666—21no—
36#4666—21no—
37#4665—21no—
38#4665—21no—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#2297—31no—
45#2285—31no—
46#2096D311yescharacteristic, complemented
47#2093—31no—
48#2072—31no—
49#2071—31no—
50#1928—32no—
51#1926—32no—
52#1854—32no—
53#1817—31no—
54#1788—32no—
55C22×C3×Q8C2311yescharacteristic
56C2×#239—22no—
57C2×#239—22no—
58C2×#239—24no—
59C2×#239—42no—
60C22×D12—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4×Dic3—21no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#892—42no—
89#892—44no—
90C42×D3—24no—
91C3×#106—42no—
92C3×#103—42no—
93C3×#95—21no—
94C3×#95—21no—
95#835—24no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#821—24no—
101(C2×D3)⋊C8—24no—
102#809—42no—
103#807—42no—
104C22×#41—31no—
105C2×#117—31no—
106C2×#117—61no—
107C22×C42—31no—
108C2×#107—32no—
109C2×#106C2×D312yes—
110C2×#103—32no—
111C2×#103C2×D312yes—
112C2×#102—32no—
113C2×#102—32no—
114C2×C22⋊C8—32no—
115#391—31no—
116#391C2×D311yescharacteristic
117#390—31no—
118#390—31no—
119#386—61no—
120#386—61no—
121#384—31no—
122#384C2×D311yescharacteristic
123#382—61no—
124#380—32no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—61no—
128C4×C22⋊C4—61no—
129C4×C22⋊C4—61no—
130#340—32no—
131#340—32no—
132#336—32no—
133#336—31no—
134#336C2×D311yescharacteristic
135C23×D3—21no—
136C23×D3C2×D411yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8#4112yes—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152#239—48no—
153#239—48no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—24no—
157C2×D12—21no—
158C2×D12—42no—
159C2×D12—42no—
160C2×D12#4112yes—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—24no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—41no—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3#4114yes—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C42—22no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—42no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—22no—
197C4×Dic3—42no—
198C4×Dic3—44no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×#41—64no—
203C2×#41—62no—
204C2×#41—62no—
205C2×#41—122no—
206C22×Q8C22×D311yescharacteristic
207C22×D4—31no—
208C22×D4—61no—
209C23×C4—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C22×C8—32no—
214C22×C8C22×D312yes—
215#117—122no—
216#117—124no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—64no—
235C2×C42—61no—
236C2×C42—62no—
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—42no—
240#103—42no—
241#103—64no—
242#102—64no—
243#102—122no—
244C22⋊C8—64no—
245#95—21no—
246#95—21no—
247#95—61no—
248#95—61no—
249#95—62no—
250#95—62no—
251C23×C3C22×D411yescharacteristic
252C22×D3—21no—
253C22×D3—21no—
254C22×D3—21no—
255C22×D3—21no—
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3#12312yes—
266C3×Q8—24no—
267C3×Q8—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4#13512yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4112yes—
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)12yes—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3#12312yes—
301C2×Dic3#13712yes—
302D12—24no—
303D12—48no—
304D12—42no—
305C4×D3—28no—
306C4×D3—24no—
307C4×D3—24no—
308C4×D3—48no—
309C4×D3—44no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—31no—
313C24—61no—
314#41—128no—
315#41—128no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8#23912yes—
320C2×D4—32no—
321C2×D4—64no—
322C2×D4—62no—
323C2×D4—61no—
324C2×D4—62no—
325C2×D4—122no—
326C2×D4—122no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—34no—
333C22×C4—31no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—64no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4—121no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4C23×D311yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—62no—
355C2×C8—62no—
356C2×C8—62no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—122no—
368C22⋊C4—122no—
369C22⋊C4—122no—
370C22⋊C4—124no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C42—22no—
374C42—62no—
375C42—62no—
376C42—62no—
377C42—122no—
378C42—124no—
379C22×C3C2×#12311yescharacteristic
380C22×C3#52511yescharacteristic
381C22×C3#53511yescharacteristic
382C22×C3C2×#13511yescharacteristic
383C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C3C2×#13711yescharacteristic
385C22×C3#57511yescharacteristic
386C2×D3—22no—
387C2×D3—22no—
388C2×D3—22no—
389C2×D3—41no—
390C2×D3—41no—
391C2×D3—41no—
392C2×D3—41no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—81no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—42no—
403C3×C4—82no—
404C3×C4—81no—
405C3×C4#55112yes—
406C3×C4#55112yes—
407Dic3—22no—
408Dic3—22no—
409Dic3—41no—
410Dic3—82no—
411Dic3—81no—
412C23—32no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23—62no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23C2×D3×D411yescharacteristic
427Q8—24no—
428Q8—42no—
429D4—64no—
430D4—128no—
431D4—122no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—68no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—124no—
458C2×C4—128no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4#91412yes—
466C2×C4#91612yes—
467C2×C4#100412yes—
468C2×C4C2×#23911yescharacteristic
469C2×C4D3×#4112yes—
470C2×C4D3×#4111yescharacteristic
471C8—42no—
472C8—122no—
473C2×C3#248211yescharacteristic
474C2×C3#309111yescharacteristic
475C2×C3#309211yescharacteristic
476C2×C3#310711yescharacteristic
477C2×C3#311511yescharacteristic
478C2×C3#320711yescharacteristic
479C2×C3#321611yescharacteristic
480D3—42no—
481D3—82no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—122no—
490C22—122no—
491C22—122no—
492C22—241no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22#562711yescharacteristic
497C22D3×#12311yescharacteristic
498C22#571511yescharacteristic
499C22C2×#91411yescharacteristic
500C22C2×#91611yescharacteristic
501C22D3×#13711yescharacteristic
502C22#577711yescharacteristic
503C4—22no—
504C4—22no—
505C4—42no—
506C4—62no—
507C4—62no—
508C4—82no—
509C4—81no—
510C4—121no—
511C4—241no—
512C4—242no—
513C4#563712yes—
514C4#565912yes—
515C3#2758711yescharacteristic, complemented, Sylow
516C2—122no—
517C2—242no—
518C2#7810511yescharacteristic
519C2#8099711yescharacteristic
520C2#8099911yescharacteristic
521C2#8142611yescharacteristic
522C2#8142711yescharacteristic
523C2#8176511yescharacteristic
524C2#8187511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2096C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#2096D3
hypocenterC3#27587
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
2. #74318C2semidirect
3. C3×#2096C2semidirect
4. #70580C2semidirect
5. #70569C2semidirect
7. #70242C2semidirect
8. #69670C2semidirect
9. #69586C2semidirect
10. #68664C2semidirect
11. #68635C2semidirect
46. #2096D3semidirect
515. C3#27587semidirect

Maximal subgroups

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

subgroupindex
#764782
#743182
C3×#20962
#705802
#705692
#702672
#702422
#696702
#695862
#686642
#686352
#275873