Finite Group Databasebeta

← #10697513

Subgroups — #10697513

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#462C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic
4#71787C211yesmaximal, characteristic, complemented
5C2×#4535C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67702C211yesmaximal, characteristic, complemented
8#10233—31nomaximal, Sylow
9C22×C3×Q16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#946—21no—
13C2×C4×D12C2211yescharacteristic
14C3×#462—22no—
15C3×#462—22no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18#5087—22no—
19#5087—22no—
20#4994—22no—
21#4987—21no—
22C2×#809—21no—
23C2×#807—21no—
24#4939—21no—
25#4935C2211yescharacteristic
26#4928—21no—
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4×C3⋊C8C2211yescharacteristic
29#4535—22no—
30#4529—22no—
31#4483—22no—
32C2×#474—31no—
33C2×#462D311yescharacteristic, complemented
34#2538—31no—
35#1990—31no—
36C2×Q8⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×Q16—24no—
44C2×C3×Q16—42no—
45C2×C3×Q16—42no—
46C22×C3×C8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61#946—42no—
62#946—42no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D412yes—
70C2×Dic3⋊C4—22no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8D412yes—
79C3×#103—24no—
80C3⋊#95—22no—
81#809—24no—
82#807—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C22×Q16—21no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×C22⋊Q8—61no—
90C2×C4×Q8C2×D311yescharacteristic
91C2×C4×D4—31no—
92#474—62no—
93#474—62no—
94#462—22no—
95#462—22no—
96C2×#107—61no—
97C2×C4⋊C8—31no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#103—21no—
100C2×#102—61no—
101C2×C4×C8—31no—
102#396—62no—
103#390—61no—
104#389—61no—
105#384—31no—
106#380—61no—
107#340—62no—
108Q8⋊C8—62no—
109D4⋊C8—62no—
110C23×D3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×Q8D812yes—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—22no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12Q1612yescomplemented
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C3×Q16—48no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161(C2×D3)⋊C4—44no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171C4⋊Dic3—42no—
172Dic3⋊C4—44no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C22×Q8—21no—
180C22×Q8#6711yescharacteristic
181C22×D4—31no—
182C23×C4—61no—
183C2×Q16—24no—
184C2×Q16—42no—
185C2×Q16—42no—
186C22×C8—21no—
187C22×C8—61no—
188C22×C8—61no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8#6712yes—
193C22⋊Q8—122no—
194C22⋊Q8—122no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4#6711yescharacteristic
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—62no—
215C2×C42—21no—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—21no—
219C4⋊C8—32no—
220C4⋊C8—61no—
221C4⋊C8#6712yes—
222#103—24no—
223#102—64no—
224C4×C8—61no—
225C4×C8—61no—
226#95—62no—
227C23×C3#12011yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—42no—
240C3×Q8—42no—
241C3×Q8—81no—
242C3×Q8—81no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12711yescharacteristic
260C2×C3×C4C2×D811yescharacteristic
261C2×C3×C4C2×Q1611yescharacteristic
262C2×C3×C4#13511yescharacteristic
263C2×C3×C4#13511yescharacteristic
264C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
265C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
266C2×C3×C4#13711yescharacteristic
267C2×C3×C4#13711yescharacteristic
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274D12—22no—
275D12—22no—
276D12—41no—
277D12—41no—
278C4×D3—81no—
279C4×D3—81no—
280C4×D3—81no—
281C4×D3—81no—
282C3×C8—42no—
283C3⋊C8—42no—
284C3⋊C8—42no—
285C24—61no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8#21312yes—
298C2×D4—32no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—61no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—61no—
308C22×C4—62no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4C2×#6711yescharacteristic
318Q16—48no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—121no—
325C2×C8—121no—
326C2×C8—121no—
327C2×C8—121no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—122no—
339C4⋊C4—124no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C22⋊C4—124no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—242no—
348C42—41no—
349C42—41no—
350C42D3×D411yescharacteristic
351C42#23711yescharacteristic
352C42C2×#6711yescharacteristic
353C42C2×#6711yescharacteristic
354C22×C3#39211yescharacteristic
355C22×C3#44711yescharacteristic
356C22×C3#44911yescharacteristic
357C22×C3#45111yescharacteristic
358C22×C3#45211yescharacteristic
359C22×C3#45511yescharacteristic
360C22×C3#45611yescharacteristic
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—42no—
371C3×C4—82no—
372C3×C4—81no—
373C3×C4#44711yescharacteristic
374C3×C4#44911yescharacteristic
375C3×C4#45111yescharacteristic
376C3×C4#45211yescharacteristic
377C3×C4#46611yescharacteristic
378C3×C4#46711yescharacteristic
379C3×C4#46911yescharacteristic
380C3×C4#47011yescharacteristic
381Dic3—82no—
382Dic3—81no—
383C23—62no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23#93711yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395Q8—42no—
396Q8—42no—
397Q8—81no—
398Q8—81no—
399D4—62no—
400D4—62no—
401D4—121no—
402D4—121no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—122no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4D3×Q1611yescharacteristic
425C2×C4#91811yescharacteristic
426C2×C4#91811yescharacteristic
427C2×C4#91911yescharacteristic
428C2×C4C2×#21311yescharacteristic
429C2×C4#93211yescharacteristic
430C2×C4#93711yescharacteristic
431C2×C4#93811yescharacteristic
432C2×C4#93911yescharacteristic
433C2×C4#93911yescharacteristic
434C2×C4#94011yescharacteristic
435C2×C4#94911yescharacteristic
436C2×C4#95011yescharacteristic
437C2×C4#95311yescharacteristic
438C8—42no—
439C8—122no—
440C8—122no—
441C2×C3#164111yescharacteristic
442C2×C3#164411yescharacteristic
443C2×C3#164911yescharacteristic
444C2×C3#165211yescharacteristic
445C2×C3#199911yescharacteristic
446C2×C3#200111yescharacteristic
447C2×C3#200711yescharacteristic
448D3—82no—
449C22—122no—
450C22—122no—
451C22—122no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—241no—
456C22#503611yescharacteristic
457C22#503711yescharacteristic
458C22#518611yescharacteristic
459C22#518711yescharacteristic
460C22#520211yescharacteristic
461C22#520511yescharacteristic
462C22#522311yescharacteristic
463C4—21no—
464C4—21no—
465C4—42no—
466C4—82no—
467C4—81no—
468C4—241no—
469C4—242no—
470C4#503811yescharacteristic
471C4#503911yescharacteristic
472C4#518611yescharacteristic
473C4#518711yescharacteristic
474C4#518811yescharacteristic
475C4#518911yescharacteristic
476C4#521711yescharacteristic
477C4#523811yescharacteristic
478C3#1023311yescharacteristic, complemented, Sylow
479C2—242no—
480C2#7106111yescharacteristic
481C2#7106211yescharacteristic
482C2#7106511yescharacteristic
483C2#7106611yescharacteristic
484C2#7168411yescharacteristic
485C2#7207111yescharacteristic
486C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#462C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#462D3
hypocenterC3#10233
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×C3×#462C2semidirect
4. #71787C2semidirect
5. C2×#4535C2semidirect
7. #67702C2semidirect
33. C2×#462D3semidirect
136. C2×D12Q16semidirect
478. C3#10233semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×#50872
#825002
#717872
C2×#45352
C2×#44832
#677022
#102333