Finite Group Databasebeta

← #10814915

Subgroups — #10814915

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#204C211yesmaximal, characteristic
2#74325C211yesmaximal, characteristic, complemented
3#74324C211yesmaximal, characteristic
4C3×#1844C211yesmaximal, characteristic, complemented
5#70574C212yesmaximal
6#70570C212yesmaximal, complemented
7#70569C212yesmaximal
8Dic3×#106C212yesmaximal, complemented
9C8×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic
12#22805—31nomaximal, Sylow
13C22×#204C2211yescharacteristic
14C2×#892—21no—
15C2×C4×D12C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C2×#821C2211yescharacteristic
23C2×#817C2211yescharacteristic
24C2×#816C2211yescharacteristic
25C2×C8×Dic3C2212yes—
26#4990—22no—
27#4988C2212yes—
28#4987—21no—
29#4978—22no—
30#4977—21no—
31Dic3×C4⋊C4—21no—
32Dic3×C4⋊C4C2212yes—
33Dic3×C4⋊C4C2212yes—
34#4973C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C4×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C4×#204C418yescomplemented
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2355—31no—
45#2354—31no—
46#1968—32no—
47#1927—32no—
48#1926—32no—
49#1844D311yescharacteristic, complemented
50#1803—31no—
51#1802—31no—
52C4×#106—32no—
53C8×C4⋊C4—31no—
54C22×D12C2311yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42C2311yescharacteristic
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3C2311yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2312yes—
78C22×C3⋊C8C2312yes—
79C2×#204—22no—
80C2×#204—22no—
81C2×#204C2×C414yes—
82#892—44no—
83C4×D12—22no—
84C4×D12C2×C414yes—
85C4×Dic6—22no—
86C4×Dic6C2×C414yes—
87C3×#106—24no—
88C3×#106—42no—
89C3×C4×C8C2×C414yes—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95D412yes—
95#821C2×C414yes—
96#817C2×C414yes—
97#816C2×C414yes—
98C8×Dic3—24no—
99#810—42no—
100#809—42no—
101C4⋊(C3⋊C8)—22no—
102C22×SD16—31no—
103C2×C4×Q8—31no—
104C2×C4×D4—31no—
105C2×#117—61no—
106C4×SD16—38no—
107C2×#106—31no—
108C2×#106C2×D312yes—
109C2×#106C2×D312yes—
110C2×C4⋊C8—31no—
111C2×#103—31no—
112C2×#103—32no—
113C2×#102—31no—
114C2×#102—32no—
115C2×C4×C8—32no—
116C2×C4×C8C2×D311yescharacteristic
117#391—32no—
118#390—32no—
119#388—62no—
120#387—62no—
121#384C2×D311yescharacteristic
122#384C2×D311yescharacteristic
123#382—61no—
124#380—61no—
125C4×C4⋊C4—31no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—32no—
128C4×C4⋊C4—61no—
129#340—32no—
130#340—32no—
131#336—32no—
132C23×D3—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3C2×D412yes—
142C22×Dic3C2×D412yes—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12C22×C412yes—
149C2×C4×D3—24no—
150C2×C4×D3—42no—
151C2×C4×D3—42no—
152C2×Dic6—21no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—21no—
156C2×Dic6—21no—
157C2×Dic6C22×C412yes—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8C22×C412yes—
161C2×C3×C8C22×C412yes—
162C2×C3×C8#4112yes—
163C2×C3×C8#4112yes—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—81no—
171C3×C4⋊C4—81no—
172C3×C42C22×C412yes—
173C3×C42#4112yes—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177C4⋊Dic3—22no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3SD1614yescomplemented
181C4⋊Dic3C22×C412yes—
182C4⋊Dic3C22×C412yes—
183Dic3⋊C4—24no—
184C4×Dic3—24no—
185C4×Dic3—22no—
186C4×Dic3—22no—
187C4×Dic3—44no—
188C4×Dic3—82no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192#204—28no—
193C22×Q8—31no—
194C22×D4—31no—
195C23×C4—61no—
196C2×SD16—34no—
197C2×SD16—62no—
198C2×SD16—62no—
199C22×C8—32no—
200C22×C8C22×D311yescharacteristic
201C22×C8C22×D311yescharacteristic
202C4×Q8—34no—
203C4×Q8—62no—
204C4×D4—34no—
205C4×D4—62no—
206#117—124no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4C22×D312yes—
217C2×C4⋊C4C22×D312yes—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—62no—
221C2×C42—32no—
222C2×C42—61no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42C22×D311yescharacteristic
226#106—24no—
227#106—34no—
228#106—42no—
229C4⋊C8—62no—
230#103—34no—
231#103—122no—
232#102—34no—
233#102—122no—
234C4×C8—64no—
235C4×C8C4×D314yes—
236#95—32no—
237#95—62no—
238#95—62no—
239#95—62no—
240#95—62no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4C4×D412yes—
258C2×C3×C4C4×D412yes—
259C2×C3×C4C2×SD1612yes—
260C2×C3×C4#13512yes—
261C2×C3×C4C23×C411yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4H(5,𝔽2)11yescharacteristic
266C2×C3×C42−1+411yescharacteristic
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3C2×SD1614yes—
282D12—22no—
283D12—22no—
284D12—41no—
285D12—41no—
286C4×D3—48no—
287Dic6—22no—
288Dic6—22no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—22no—
292C3×C8#11714yescomplemented
293C3⋊C8—42no—
294C24—61no—
295C2×Q8—32no—
296C2×Q8—61no—
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—61no—
300C2×Q8—61no—
301C2×D4—32no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—64no—
315C22×C4—122no—
316C22×C4—122no—
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4C23×D311yescharacteristic
320SD16—68no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8C2×C4×D312yes—
327C2×C8C2×C4×D312yes—
328C2×C8#23712yes—
329C2×C8#23912yes—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—34no—
335C4⋊C4—32no—
336C4⋊C4—32no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—64no—
340C4⋊C4—62no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C22⋊C4—64no—
346C22⋊C4—124no—
347C22⋊C4—242no—
348C42—64no—
349C42—62no—
350C42—62no—
351C42—124no—
352C42—242no—
353C42C2×C4×D312yes—
354C42#23912yes—
355C22×C3C2×C4×D411yescharacteristic
356C22×C3#54711yescharacteristic
357C22×C3#55011yescharacteristic
358C22×C3C22×SD1611yescharacteristic
359C22×C3C2×#13511yescharacteristic
360C22×C3#57711yescharacteristic
361C22×C3#57711yescharacteristic
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—81no—
369C3×C4—22no—
370C3×C4—82no—
371C3×C4—82no—
372C3×C4C4×SD1614yes—
373C3×C4C2×#11711yescharacteristic
374C3×C4#52011yescharacteristic
375C3×C4#55511yescharacteristic
376C3×C4#55611yescharacteristic
377Dic3—24no—
378Dic3—42no—
379Dic3—42no—
380Dic3—42no—
381Dic3—42no—
382C23—62no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23C2×D3×D411yescharacteristic
390Q8—62no—
391Q8—62no—
392Q8—121no—
393Q8—121no—
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—34no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—128no—
422C2×C4#89312yes—
423C2×C4#89312yes—
424C2×C4D3×SD1612yes—
425C2×C4#91612yes—
426C2×C4C22×C4×D311yescharacteristic
427C2×C4C2×#23711yescharacteristic
428C2×C4#100411yescharacteristic
429C2×C4C2×#23911yescharacteristic
430C2×C4C2×#23911yescharacteristic
431C2×C4#100711yescharacteristic
432C8—22no—
433C8—122no—
434C8#89214yescomplemented
435C2×C3#235411yescharacteristic
436C2×C3C2×C4×SD1611yescharacteristic
437C2×C3#295211yescharacteristic
438C2×C3#333511yescharacteristic
439C2×C3#334011yescharacteristic
440C2×C3#339211yescharacteristic
441C2×C3#339411yescharacteristic
442D3—82no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22—241no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C2×#89311yescharacteristic
451C22#571811yescharacteristic
452C22#572611yescharacteristic
453C22C2×D3×SD1611yescharacteristic
454C22C2×#91611yescharacteristic
455C22#576211yescharacteristic
456C22#576211yescharacteristic
457C4—22no—
458C4—64no—
459C4—82no—
460C4—82no—
461C4—122no—
462C4—122no—
463C4—122no—
464C4—122no—
465C4#485214yes—
466C4C2×#89211yescharacteristic
467C4#550411yescharacteristic
468C4#573211yescharacteristic
469C4#573411yescharacteristic
470C3#2280511yescharacteristic, complemented, Sylow
471C2—242no—
472C2#7432411yescharacteristic
473C2C2×#485211yescharacteristic
474C2#7910611yescharacteristic
475C2#8175511yescharacteristic
476C2#8175611yescharacteristic
477C2#8189911yescharacteristic
478C2#8190011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1844C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1844D3
hypocenterC3#22805
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. #74325C2semidirect
4. C3×#1844C2semidirect
6. #70570C2semidirect
8. Dic3×#106C2semidirect
9. C8×C4⋊Dic3C2semidirect
10. #69588C2semidirect
39. C4×#204C4semidirect
49. #1844D3semidirect
180. C4⋊Dic3SD16semidirect
292. C3×C8#117semidirect
434. C8#892semidirect
470. C3#22805semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2042
#743252
#743242
C3×#18442
#705742
#705702
#705692
Dic3×#1062
C8×C4⋊Dic32
#695882
#695842
#228053