Finite Group Databasebeta

← #10814355

Subgroups — #10814355

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#74324C211yesmaximal, characteristic
4C3×#1803C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic
6#70466C211yesmaximal, characteristic, complemented
7#70240C212yesmaximal
8#70220C212yesmaximal, complemented
9#69574C211yesmaximal, characteristic, complemented
10#68857C212yesmaximal
11#68817C212yesmaximal, complemented
12#22759—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C22×#215C2211yescharacteristic
16C2×#892—21no—
17C2×C4×D12C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#103C2212yes—
20C3×#384C2211yescharacteristic
21C2×#835C2211yescharacteristic
22C2×#821C2212yes—
23C2×#815C2212yes—
24C4×#215C418yescomplemented
25#4990—22no—
26#4988C2212yes—
27#4987—21no—
28#4984—22no—
29#4984C2212yes—
30Dic3×C4⋊C4C2212yes—
31C2×#809C2211yescharacteristic
32C2×#808C2211yescharacteristic
33#4935—21no—
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38C3×#340C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2354—31no—
45Q8×C4⋊C4—31no—
46#1936—32no—
47#1919—32no—
48#1880—32no—
49#1834—31no—
50#1803D311yescharacteristic, complemented
51#1802—31no—
52#1770—31no—
53#1763—32no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C3×C8C2312yes—
58C3×C4×Q8—22no—
59C3×C4×Q8C2×C414yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67Q8×Dic3—24no—
68Q8×Dic3—42no—
69C2×#215—22no—
70C2×#215—22no—
71C2×#215C2×C414yes—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3D411yescharacteristic
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3C2312yes—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88#892—44no—
89C4×D12—22no—
90C4×D12C2×C414yes—
91C3×C4⋊C8—22no—
92C3×#103—24no—
93#835C2×C414yes—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95D412yes—
97#821—42no—
98#815—42no—
99#809C2×C414yes—
100#808C2×C414yes—
101C4×C3⋊C8C2×C414yes—
102C22×SD16—31no—
103C2×C4×Q8—32no—
104C2×C4×Q8C2×D311yescharacteristic
105C2×C4×D4—31no—
106C2×#117—61no—
107C4×SD16—38no—
108C2×#106—31no—
109C2×C4⋊C8C2×D311yescharacteristic
110C2×#103—31no—
111C2×#103C2×D312yes—
112C2×#102—31no—
113C2×#102—32no—
114C2×#97—32no—
115C2×C4×C8—31no—
116#390—32no—
117#389—32no—
118#389—62no—
119#388—62no—
120#384—31no—
121#384—61no—
122#384C2×D311yescharacteristic
123#380—61no—
124C4×C4⋊C4—31no—
125C4×C4⋊C4—32no—
126C4×C4⋊C4—61no—
127#340—32no—
128#340—32no—
129#340—32no—
130#340C2×D312yes—
131#336—32no—
132C23×D3—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8C22×C412yes—
137C2×C3×Q8#4112yes—
138C2×C3×Q8#4112yes—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3C2×D412yes—
148C22×Dic3C2×D412yes—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12C22×C412yes—
155C2×C4×D3—24no—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—81no—
167C3×C4⋊C4C22×C412yes—
168C3×C42—24no—
169C3×C42C22×C412yes—
170C3×C42#4112yes—
171#215—28no—
172(C2×D3)⋊C4—24no—
173(C2×D3)⋊C4—44no—
174(C2×D3)⋊C4—82no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—44no—
181C4⋊Dic3—42no—
182C4⋊Dic3—82no—
183C4⋊Dic3—81no—
184C4⋊Dic3C22×C412yes—
185C4×Dic3—22no—
186C4×Dic3—22no—
187C4×Dic3—44no—
188C4×Dic3—44no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8C22×C412yes—
194C2×C3⋊C8C22×C412yes—
195C22×Q8C22×D311yescharacteristic
196C22×D4—31no—
197C23×C4—61no—
198C2×SD16—34no—
199C2×SD16—62no—
200C2×SD16—62no—
201C22×C8—31no—
202C22×C8—31no—
203C22×C8C22×D312yes—
204C4×Q8—22no—
205C4×Q8—64no—
206C4×Q8—122no—
207C4×Q8C4×D314yes—
208C4×D4—34no—
209C4×D4—62no—
210#117—124no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C22⋊C4—61no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C42—21no—
228C2×C42—32no—
229C2×C42—62no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#106—34no—
233C4⋊C8—22no—
234#103—24no—
235#103—34no—
236#102—34no—
237#102—122no—
238#97—122no—
239C4×C8—34no—
240#95—32no—
241#95—62no—
242#95—62no—
243C23×C3C22×D411yescharacteristic
244C22×D3—22no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C3×Q8—22no—
252C3×Q8—22no—
253C3×Q8—22no—
254C3×Q8#11714yescomplemented
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4C4×D412yes—
267C2×C3×C4C4×D412yes—
268C2×C3×C4AGL(1,ℤ/8)12yes—
269C2×C3×C4#13712yes—
270C2×C3×C4C23×C411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4H(5,𝔽2)11yescharacteristic
275C2×C3×C42−1+411yescharacteristic
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288D12—22no—
289D12—22no—
290D12—41no—
291D12—41no—
292C4×D3—48no—
293C3×C8—42no—
294C3⋊C8—22no—
295C3⋊C8—22no—
296C24—61no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C2×C4×D312yes—
301C2×Q8#23712yes—
302C2×Q8#23912yes—
303C2×D4—32no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—61no—
315C22×C4—64no—
316C22×C4—62no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4C23×D311yescharacteristic
322SD16—68no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—32no—
327C2×C8—32no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—61no—
331C2×C8—61no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—32no—
337C4⋊C4—42no—
338C4⋊C4—62no—
339C4⋊C4—62no—
340C4⋊C4—61no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—81no—
344C4⋊C4—124no—
345C4⋊C4—122no—
346C4⋊C4—241no—
347C4⋊C4—242no—
348C4⋊C4C2×C4×D312yes—
349C22⋊C4—64no—
350C22⋊C4—124no—
351C22⋊C4—242no—
352C42—24no—
353C42—62no—
354C42—62no—
355C42—124no—
356C42—124no—
357C42C2×C4×D312yes—
358C42#23712yes—
359C22×C3C2×C4×D411yescharacteristic
360C22×C3#54611yescharacteristic
361C22×C3#54811yescharacteristic
362C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
363C22×C3C2×#13711yescharacteristic
364C22×C3#57711yescharacteristic
365C22×C3#57711yescharacteristic
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—42no—
369C2×D3—81no—
370C2×D3—81no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—24no—
374C3×C4—22no—
375C3×C4—42no—
376C3×C4—82no—
377C3×C4#44014yes—
378C3×C4C2×#11711yescharacteristic
379C3×C4#52011yescharacteristic
380C3×C4#55211yescharacteristic
381C3×C4#55311yescharacteristic
382Dic3—42no—
383Dic3—42no—
384Dic3—42no—
385Dic3—82no—
386C23—62no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23C2×D3×D411yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—22no—
397Q8#89214yescomplemented
398D4—62no—
399D4—62no—
400D4—121no—
401D4—121no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—128no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4#67⋊C412yes—
427C2×C4#67⋊C412yes—
428C2×C4#91412yes—
429C2×C4#91812yes—
430C2×C4C22×C4×D311yescharacteristic
431C2×C4C2×#23711yescharacteristic
432C2×C4C2×#23711yescharacteristic
433C2×C4C2×#23911yescharacteristic
434C2×C4#100911yescharacteristic
435C2×C4#101011yescharacteristic
436C8—42no—
437C8—62no—
438C8—62no—
439C2×C3#235211yescharacteristic
440C2×C3C2×#44011yescharacteristic
441C2×C3#295211yescharacteristic
442C2×C3#332911yescharacteristic
443C2×C3#333211yescharacteristic
444C2×C3#336811yescharacteristic
445C2×C3#336911yescharacteristic
446D3—82no—
447C22—122no—
448C22—122no—
449C22—122no—
450C22—241no—
451C22—241no—
452C22—241no—
453C22—241no—
454C22C2×#67⋊C411yescharacteristic
455C22#559811yescharacteristic
456C22#563011yescharacteristic
457C22C2×#91411yescharacteristic
458C22C2×#91811yescharacteristic
459C22#577511yescharacteristic
460C22#577811yescharacteristic
461C4—22no—
462C4—24no—
463C4—42no—
464C4—82no—
465C4—122no—
466C4—122no—
467C4—122no—
468C4—242no—
469C4#481714yes—
470C4C2×#89211yescharacteristic
471C4#552311yescharacteristic
472C4#561811yescharacteristic
473C4#564211yescharacteristic
474C3#2275911yescharacteristic, complemented, Sylow
475C2—242no—
476C2#7388311yescharacteristic
477C2C2×#481711yescharacteristic
478C2#7886911yescharacteristic
479C2#8037111yescharacteristic
480C2#8046411yescharacteristic
481C2#8102711yescharacteristic
482C2#8112311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1803D3
hypocenterC3#22759
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic3C2semidirect
4. C3×#1803C2semidirect
6. #70466C2semidirect
8. #70220C2semidirect
9. #69574C2semidirect
11. #68817C2semidirect
24. C4×#215C4semidirect
50. #1803D3semidirect
254. C3×Q8#117semidirect
397. Q8#892semidirect
474. C3#22759semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic32
C2×C4×#2152
#743242
C3×#18032
#704792
#704662
#702402
#702202
#695742
#688572
#688172
#227593