Finite Group Databasebeta

← #10814350

Subgroups — #10814350

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
1C2×C4×#204C211yesmaximal, characteristic
2#74324C211yesmaximal, characteristic
3#74323C211yesmaximal, characteristic, complemented
4C3×#1834C211yesmaximal, characteristic, complemented
5#69757C211yesmaximal, characteristic, complemented
6#69588C211yesmaximal, characteristic, complemented
7#69584C211yesmaximal, characteristic
8#68909C212yesmaximal
9#68856C212yesmaximal
10#68816C212yesmaximal, complemented
11#68783C212yesmaximal, complemented
12#22759—31nomaximal, Sylow
13C22×#204C2211yescharacteristic
14C2×#892—21no—
15C2×#887C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×C4×C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×#817C2211yescharacteristic
23C2×#816C2211yescharacteristic
24C2×#815C2212yes—
25#4990—22no—
26#4988C2212yes—
27#4987—21no—
28#4976—22no—
29#4975—21no—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4C2212yes—
32#4972C2212yes—
33C2×#810C2212yes—
34C2×#809C2212yes—
35C4×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C4×#204C418yescomplemented
38C3×#340C2212yes—
39C3×#340C2212yes—
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#1834D311yescharacteristic, complemented
50#1803—31no—
51#1802—31no—
52#1770—31no—
53#1763—32no—
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×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3C2312yes—
80C22×C3⋊C8C2312yes—
81C2×#204—22no—
82C2×#204—22no—
83C2×#204C2×C414yes—
84#892—44no—
85#887—24no—
86#887—42no—
87C4×D12—22no—
88C4×D12C2×C414yes—
89C4×Dic6—22no—
90C4×Dic6C2×C414yes—
91C3×C4×C8C2×C414yes—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95D412yes—
95#821C2×C414yes—
96#817C2×C414yes—
97#816C2×C414yes—
98#815—42no—
99#810—24no—
100#809—42no—
101C4⋊(C3⋊C8)—22no—
102C22×SD16—31no—
103C2×C4×Q8—32no—
104C2×C4×Q8—31no—
105C2×C4×D4—31no—
106C2×#117—61no—
107C4×SD16—38no—
108C2×#106—31no—
109C2×C4⋊C8—31no—
110C2×#103—31no—
111C2×#103—32no—
112C2×#102—31no—
113C2×#102—32no—
114C2×#97—32no—
115C2×C4×C8C2×D311yescharacteristic
116#390—32no—
117#389—32no—
118#389—62no—
119#388—62no—
120#384—61no—
121#384C2×D311yescharacteristic
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#340C2×D312yes—
130#340C2×D312yes—
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×Dic6C22×C412yes—
156C2×Dic6#4112yes—
157C2×Dic6#4112yes—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8C22×C412yes—
163C2×C3×C8C22×C412yes—
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—22no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3C22×C412yes—
182C4⋊Dic3C22×C412yes—
183Dic3⋊C4—24no—
184Dic3⋊C4—44no—
185Dic3⋊C4—82no—
186C4×Dic3—24no—
187C4×Dic3—22no—
188C4×Dic3—22no—
189C4×Dic3—44no—
190C4×Dic3—44no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194#204—28no—
195C22×Q8—31no—
196C22×D4—31no—
197C23×C4—61no—
198C2×SD16—34no—
199C2×SD16—62no—
200C2×SD16—62no—
201C22×C8—32no—
202C22×C8C22×D311yescharacteristic
203C22×C8C22×D311yescharacteristic
204C4×Q8—34no—
205C4×Q8—62no—
206C4×Q8—64no—
207C4×Q8—122no—
208C4×D4—34no—
209C4×D4—62no—
210#117—124no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D312yes—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C42—32no—
228C2×C42—61no—
229C2×C42—62no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#106—34no—
233C4⋊C8—62no—
234#103—34no—
235#103—64no—
236#102—34no—
237#102—122no—
238#97—122no—
239C4×C8C4×D314yes—
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—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4C4×D412yes—
260C2×C3×C4C4×D412yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262C2×C3×C4#13712yes—
263C2×C3×C4C23×C411yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4H(5,𝔽2)11yescharacteristic
268C2×C3×C42−1+411yescharacteristic
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284D12—22no—
285D12—22no—
286D12—41no—
287D12—41no—
288C4×D3—48no—
289Dic6—22no—
290Dic6—22no—
291Dic6—22no—
292Dic6#11714yescomplemented
293C3×C8—22no—
294C3×C8—22no—
295C3⋊C8—42no—
296C24—61no—
297C2×Q8—32no—
298C2×Q8—32no—
299C2×Q8—32no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×D4—32no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—32no—
312C22×C4—32no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—64no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4C23×D311yescharacteristic
322SD16—68no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—62no—
330C2×C8C2×C4×D312yes—
331C2×C8C2×C4×D312yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—64no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—124no—
348C4⋊C4—242no—
349C22⋊C4—64no—
350C22⋊C4—124no—
351C22⋊C4—242no—
352C42—64no—
353C42—62no—
354C42—62no—
355C42—124no—
356C42—124no—
357C42C2×C4×D312yes—
358C42#23912yes—
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—22no—
374C3×C4—82no—
375C3×C4—82no—
376C3×C4#44014yes—
377C3×C4C2×#11711yescharacteristic
378C3×C4#52011yescharacteristic
379C3×C4#55211yescharacteristic
380C3×C4#55311yescharacteristic
381Dic3—24no—
382Dic3—42no—
383Dic3—42no—
384Dic3—42no—
385Dic3—42no—
386C23—62no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23C2×D3×D411yescharacteristic
394Q8—34no—
395Q8—62no—
396Q8—62no—
397Q8—62no—
398D4—62no—
399D4—62no—
400D4—121no—
401D4—121no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—128no—
426C2×C4#89312yes—
427C2×C4#89312yes—
428C2×C4#91112yes—
429C2×C4#91812yes—
430C2×C4C22×C4×D311yescharacteristic
431C2×C4C2×#23911yescharacteristic
432C2×C4D3×#4111yescharacteristic
433C2×C4D3×#4111yescharacteristic
434C2×C4#100911yescharacteristic
435C2×C4#101011yescharacteristic
436C8—22no—
437C8—22no—
438C8—122no—
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×#89311yescharacteristic
455C22#567611yescharacteristic
456C22#567711yescharacteristic
457C22C2×#91111yescharacteristic
458C22#576211yescharacteristic
459C22#576211yescharacteristic
460C22C2×#91811yescharacteristic
461C4—22no—
462C4—64no—
463C4—82no—
464C4—82no—
465C4—122no—
466C4—122no—
467C4—122no—
468C4—122no—
469C4#488914yes—
470C4D3×#11711yescharacteristic
471C4#552311yescharacteristic
472C4#569411yescharacteristic
473C4#569511yescharacteristic
474C3#2275911yescharacteristic, complemented, Sylow
475C2—242no—
476C2#7422911yescharacteristic
477C2#7910611yescharacteristic
478C2C2×#488911yescharacteristic
479C2#8158211yescharacteristic
480C2#8159011yescharacteristic
481C2#8161711yescharacteristic
482C2#8162611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1834C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1834D3
hypocenterC3#22759
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
3. #74323C2semidirect
4. C3×#1834C2semidirect
5. #69757C2semidirect
6. #69588C2semidirect
10. #68816C2semidirect
11. #68783C2semidirect
37. C4×#204C4semidirect
49. #1834D3semidirect
292. Dic6#117semidirect
474. C3#22759semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2042
#743242
#743232
C3×#18342
#697572
#695882
#695842
#689092
#688562
#688162
#687832
#227593