Finite Group Databasebeta

← #10691664

Subgroups — #10691664

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3190C211yesmaximal, characteristic
2#82920C211yesmaximal, characteristic
3#71708C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic, complemented
5C2×#4538C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9263—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C3×#485—21no—
12C3×#483C411yescharacteristic, complemented
13C3×#480C411yescharacteristic, complemented
14C3×#467—21no—
15C3×#465—21no—
16C3×#452—21no—
17C3×#451C411yescharacteristic, complemented
18C3×#448C411yescharacteristic, complemented
19C2×C3×C22⋊C8C2211yescharacteristic
20#5234—21no—
21C2×C3⋊#95C2211yescharacteristic
22#5207C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5011—21no—
26#5011—21no—
27D4×C3⋊C8—21no—
28D4×C3⋊C8—21no—
29#5000—21no—
30#4538—21no—
31#4538—21no—
32#4538—21no—
33#4537—21no—
34#4528—21no—
35#4528—21no—
36#3190D311yescharacteristic
37#2977—31no—
38#1504—31no—
39C2×#328—31no—
40#1472—31no—
41#1298—31no—
42#1280—31no—
43C2×C3×#41D411yescharacteristic
44C22×C3×Q8C2×C411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C2×C3×Q16—21no—
48C2×C3×SD16—21no—
49C22×C3×C8—21no—
50C3×#123—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8C2×C411yescharacteristic
56C3×C22⋊Q8D411yescharacteristic
57C3×#121C2×C411yescharacteristic
58C3×#121D411yescharacteristic
59C3×#120—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×#117D411yescharacteristic
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2×C411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×#208—21no—
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C3×#107—21no—
72C3×#106—21no—
73C3×#103—21no—
74C3×#103—21no—
75C3×#103—21no—
76C3×#102—21no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8C2×C411yescharacteristic
79C3×C22⋊C8C2×C411yescharacteristic
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830D411yescharacteristic
89#830D411yescharacteristic
90#830D411yescharacteristic
91#830D411yescharacteristic
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96#525C2×D311yescharacteristic
97C2×C22⋊Q8C2×D311yescharacteristic
98#485—21no—
99#483Dic311yescharacteristic, complemented
100#480Dic311yescharacteristic, complemented
101#467—21no—
102#465—21no—
103#452—21no—
104#451Dic311yescharacteristic, complemented
105#448Dic311yescharacteristic, complemented
106#435—61no—
107#435—61no—
108C8×D4—61no—
109C8×D4—61no—
110#433—61no—
111#408—61no—
112#407—31no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×#95—31no—
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121#324—61no—
122#324—61no—
123C24×C3#4111yescharacteristic
124C3×#41—81no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—81no—
128C2×C3×Q8C22⋊C411yescharacteristic
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4C22⋊C411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—81no—
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—81no—
147C22×Dic3—81no—
148C3×Q16—81no—
149C3×SD16—81no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—81no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4C22⋊C411yescharacteristic
160C3×C4⋊C4C22⋊C411yescharacteristic
161C3×C22⋊C4—21no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—21no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C42—21no—
169#208—81no—
170#208—81no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—81no—
177C2×C3⋊C8—81no—
178C2×C3⋊C8—81no—
179C2×#41#6711yescharacteristic
180C22×Q8C2×Dic311yescharacteristic
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C2×Q16—21no—
184C2×SD16—21no—
185C2×M16—61no—
186C2×M16—61no—
187C22×C8—21no—
188C22×C8—61no—
189C22×C8—61no—
190C22×C8—61no—
191#123—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C2×Dic311yescharacteristic
197C22⋊Q8#6711yescharacteristic
198#121C2×Dic311yescharacteristic
199#121#6711yescharacteristic
200#120—21no—
201C4×D4—21no—
202C4×D4—21no—
203#117#6711yescharacteristic
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic311yescharacteristic
206C2×C22⋊C4—21no—
207#107—21no—
208#106—21no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—21no—
212#103—21no—
213#103—21no—
214#102—21no—
215C22⋊C8—21no—
216C22⋊C8—31no—
217C22⋊C8—31no—
218C22⋊C8—31no—
219C22⋊C8—31no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8C2×Dic311yescharacteristic
227C22⋊C8C2×Dic311yescharacteristic
228#97—61no—
229C4×C8—61no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—41no—
238C23×C3C4×D411yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3#12311yescharacteristic
241C3×Q8—41no—
242C3×Q8—41no—
243C3×Q8—41no—
244C3×Q8—81no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—81no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
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—41no—
265C2×C3×C4—41no—
266C2×C3×C4—81no—
267C2×C3×C4C2×C22⋊C411yescharacteristic
268C2×C3×C4C4×D411yescharacteristic
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×Dic3—42no—
272C2×Dic3—81no—
273C2×Dic3—82no—
274C2×Dic3—81no—
275C2×Dic3—82no—
276C3×C8—41no—
277C3×C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C3⋊C8—81no—
283C24#23711yescharacteristic
284#41—81no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—81no—
288C2×Q8#21711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4#21711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—62no—
297C22×C4—81no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—241no—
301C22×C4—241no—
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4#23711yescharacteristic
305C22×C4C22×Dic311yescharacteristic
306C22×C4C2×#6711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—81no—
309SD16—81no—
310M16—241no—
311M16—241no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—81no—
319C2×C8—121no—
320C2×C8—241no—
321C2×C8—241no—
322C2×C8—241no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4#21711yescharacteristic
330C4⋊C4#21711yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3#38611yescharacteristic
354C22×C3#44311yescharacteristic
355C22×C3#44411yescharacteristic
356C22×C3#47111yescharacteristic
357C22×C3#47311yescharacteristic
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—81no—
366C3×C4—81no—
367Dic3—82no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—41no—
374C23D4×Dic311yescharacteristic
375C23#93511yescharacteristic
376C23#93911yescharacteristic
377Q8—41no—
378Q8—41no—
379Q8—41no—
380Q8—81no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—81no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—122no—
404C2×C4—241no—
405C2×C4—242no—
406C2×C4—241no—
407C2×C4—242no—
408C2×C4D4×Dic311yescharacteristic
409C2×C4#93711yescharacteristic
410C2×C4#93911yescharacteristic
411C2×C4C2×#21711yescharacteristic
412C8—41no—
413C8—41no—
414C8—121no—
415C8—121no—
416C8—121no—
417C8—121no—
418C8—241no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—81no—
425C2×C3#187611yescharacteristic
426C2×C3#190811yescharacteristic
427C2×C3#191311yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#519611yescharacteristic
443C22#519711yescharacteristic
444C22#520311yescharacteristic
445C22#520411yescharacteristic
446C22#522011yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—242no—
457C3#926311yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2#7184811yescharacteristic
465C2#7208711yescharacteristic
466C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3190C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#3190D3
hypocenterC3#9263
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3190

Complements

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

NH ≅ G/Ntype
4. #71691C2semidirect
5. C2×#4538C2semidirect
6. #67680C2semidirect
7. #67674C2semidirect
12. C3×#483C4semidirect
13. C3×#480C4semidirect
17. C3×#451C4semidirect
18. C3×#448C4semidirect
99. #483Dic3semidirect
100. #480Dic3semidirect
104. #451Dic3semidirect
105. #448Dic3semidirect
457. C3#9263semidirect

Maximal subgroups

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

subgroupindex
C3×#31902
#829202
#717082
#716912
C2×#45382
#676802
#676742
#92633