Finite Group Databasebeta

← #10698531

Subgroups — #10698531

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73379C211yesmaximal, characteristic
2C2×#4467C211yesmaximal, characteristic, complemented
3C2×#4470C211yesmaximal, characteristic
4#69932C211yesmaximal, characteristic
5#69930C211yesmaximal, characteristic, complemented
6#69420C211yesmaximal, characteristic, complemented
7C3×#1280C211yesmaximal, characteristic, complemented
8#10405—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C22×Dic3⋊C4—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5213—21no—
17#5213—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#821—21no—
21C2×#816—21no—
22#4674—21no—
23#4673—21no—
24#4672—21no—
25#4659—21no—
26#4551—21no—
27#4551—21no—
28#4470—22no—
29#4467—22no—
30#2639—31no—
31#1874—31no—
32#1872—31no—
33C2×#328—31no—
34C2×#327—31no—
35#1444—31no—
36#1280D311yescharacteristic, complemented
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C23×Dic3—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×#217—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—42no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—42no—
68C2×Dic3⋊C4—42no—
69C2×Dic3⋊C4—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82C3⋊#95D412yes—
83#821—42no—
84#816—42no—
85C2×C22⋊Q8—31no—
86C2×#121—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—61no—
89C2×#103—61no—
90C2×#102—61no—
91C2×C22⋊C8C2×D311yescharacteristic
92C2×C22⋊C8C2×D311yescharacteristic
93#400—61no—
94#397—61no—
95#394—61no—
96#393—61no—
97#386—61no—
98#386—61no—
99C2×#95—31no—
100C2×#95—61no—
101C2×#95C2×D311yescharacteristic
102#340—61no—
103#340—61no—
104#328—62no—
105#327—62no—
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—41no—
152C2×D12—41no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C2×C3×C8—41no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163#217—41no—
164#217—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—81no—
168(C2×D3)⋊C4—81no—
169(C2×D3)⋊C4—81no—
170(C2×D3)⋊C4—81no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—41no—
178C4⋊Dic3—41no—
179Dic3⋊C4—48no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182C22×Q8—31no—
183C22×D4—31no—
184C22×D4—61no—
185C23×C4—61no—
186C23×C4—61no—
187C23×C4D1211yescharacteristic
188C23×C4D1211yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—21no—
191C22×C8—21no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—122no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4—122no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215#103—122no—
216#102—122no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221#95—21no—
222#95—21no—
223#95—32no—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229#95—122no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3#12011yescharacteristic
235C23×C3#12111yescharacteristic
236C23×C3#12411yescharacteristic
237C23×C3#13511yescharacteristic
238C23×C3#13511yescharacteristic
239C23×C3AGL(1,ℤ/8)11yescharacteristic
240C23×C3#13711yescharacteristic
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12311yescharacteristic
269C2×C3×C4#12311yescharacteristic
270#67—81no—
271#67—81no—
272#67—81no—
273#67—81no—
274C2×Dic3—24no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297D12—42no—
298D12—81no—
299Dic6—42no—
300Dic6—81no—
301C3×C8—81no—
302C3×C8—81no—
303C24—61no—
304C24C2×D1211yescharacteristic
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—121no—
308C2×Q8—121no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—121no—
347C22×C4—121no—
348C22×C4#23511yescharacteristic
349C22×C4#23511yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4#23911yescharacteristic
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—128no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—241no—
376C22⋊C4—241no—
377C22⋊C4—241no—
378C22⋊C4—241no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3#39411yescharacteristic
392C22×C3#44911yescharacteristic
393C22×C3#45211yescharacteristic
394C22×C3#45311yescharacteristic
395C22×C3#45611yescharacteristic
396C22×C3B(4,𝔽2)11yescharacteristic
397C22×C3#45811yescharacteristic
398C22×C3#46311yescharacteristic
399C22×C3#46411yescharacteristic
400C22×C3#48811yescharacteristic
401C22×C3#48911yescharacteristic
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—81no—
414C3×C4—81no—
415Dic3—44no—
416Dic3—82no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23#87511yescharacteristic
432C23#88211yescharacteristic
433C23#89511yescharacteristic
434C23#90411yescharacteristic
435C23#90411yescharacteristic
436C23#90811yescharacteristic
437C23#90911yescharacteristic
438Q8—122no—
439Q8—241no—
440D4—122no—
441D4—241no—
442D4—241no—
443D4—241no—
444D4—241no—
445D4—241no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—64no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4#88411yescharacteristic
488C2×C4#88411yescharacteristic
489C2×C4#89411yescharacteristic
490C2×C4#89411yescharacteristic
491C8—81no—
492C8—81no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3#160011yescharacteristic
498C2×C3#160311yescharacteristic
499C2×C3#160911yescharacteristic
500C2×C3#161811yescharacteristic
501C2×C3#203911yescharacteristic
502C2×C3#204211yescharacteristic
503C2×C3#204411yescharacteristic
504D3—161no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22#467211yescharacteristic
525C22#469711yescharacteristic
526C22#471411yescharacteristic
527C22#473311yescharacteristic
528C22#473411yescharacteristic
529C22#474111yescharacteristic
530C22#474211yescharacteristic
531C22#474711yescharacteristic
532C22#475211yescharacteristic
533C22#484411yescharacteristic
534C22#484511yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—81no—
540C4—81no—
541C4—124no—
542C4—242no—
543C4—241no—
544C4—241no—
545C4—241no—
546C3#1040511yescharacteristic, complemented, Sylow
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—481no—
552C2#6810711yescharacteristic
553C2#6811311yescharacteristic
554C2#6813211yescharacteristic
555C2#6813411yescharacteristic
556C2#6863011yescharacteristic
557C2#6895711yescharacteristic
558C2#6897611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenter#1280D3
hypocenterC3#10405
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
2. C2×#4467C2semidirect
5. #69930C2semidirect
6. #69420C2semidirect
7. C3×#1280C2semidirect
36. #1280D3semidirect
546. C3#10405semidirect

Maximal subgroups

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

subgroupindex
#733792
C2×#44672
C2×#44702
#699322
#699302
#694202
C3×#12802
#104053