Finite Group Databasebeta

← #10691671

Subgroups — #10691671

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3188C211yesmaximal, characteristic
2#82920C211yesmaximal, characteristic, complemented
3#71708C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9264—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C3×#484C411yescharacteristic, complemented
12C3×#482—21no—
13C3×#481C411yescharacteristic, complemented
14C3×#466—21no—
15C3×#465—21no—
16C3×#450C411yescharacteristic, complemented
17C3×#449—21no—
18C3×#447C411yescharacteristic, 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#4537—21no—
32#4537—21no—
33#4537—21no—
34#4525—21no—
35#4525—21no—
36#3188D311yescharacteristic
37#2977—31no—
38#1504—31no—
39C2×#327—31no—
40#1471—31no—
41#1298—31no—
42#1280—31no—
43C2×C3×#41D411yescharacteristic
44C22×C3×D4—21no—
45C22×C3×D4C2×C411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3—21no—
48C2×C3×SD16—21no—
49C2×C3×D8—21no—
50C22×C3×C8—21no—
51C3×#123—21no—
52C3×C22⋊Q8C2×C411yescharacteristic
53C3×C22⋊Q8D411yescharacteristic
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121C2×C411yescharacteristic
59C3×#121D411yescharacteristic
60C3×#120—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×#117D411yescharacteristic
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×#102—21no—
75C3×#102—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×#121C2×D311yescharacteristic
98#484Dic311yescharacteristic, complemented
99#482—21no—
100#481Dic311yescharacteristic, complemented
101#466—21no—
102#465—21no—
103#450Dic311yescharacteristic, complemented
104#449—21no—
105#447Dic311yescharacteristic, 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#327—61no—
119#327—61no—
120#327—61no—
121#323—61no—
122#323—61no—
123C24×C3—21no—
124C24×C3#4111yescharacteristic
125C3×#41—81no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—81no—
135C2×C3×D4—81no—
136C2×C3×D4—81no—
137C2×C3×D4C22⋊C411yescharacteristic
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—81no—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—81no—
151C22×Dic3—81no—
152C3×SD16—81no—
153C3×D8—81no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—81no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4C22⋊C411yescharacteristic
162C3×C4⋊C4C22⋊C411yescharacteristic
163C3×C22⋊C4—21no—
164C3×C22⋊C4—21no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—21no—
167C3×C22⋊C4—21no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C42—21no—
171#208—81no—
172#208—81no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—81no—
179C2×C3⋊C8—81no—
180C2×C3⋊C8—81no—
181C2×#41#6711yescharacteristic
182C22×D4—21no—
183C22×D4C2×Dic311yescharacteristic
184C23×C4—61no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—21no—
187C2×D8—21no—
188C2×M16—61no—
189C2×M16—61no—
190C22×C8—21no—
191C22×C8—61no—
192C22×C8—61no—
193C22×C8—61no—
194#123—21no—
195C22⋊Q8C2×Dic311yescharacteristic
196C22⋊Q8#6711yescharacteristic
197#121—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121C2×Dic311yescharacteristic
202#121#6711yescharacteristic
203#120—21no—
204C4×D4—21no—
205C4×D4—21no—
206#117#6711yescharacteristic
207C2×C4⋊C4C2×Dic311yescharacteristic
208C2×C22⋊C4—21no—
209#107—21no—
210#106—21no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213#103—21no—
214#102—21no—
215#102—21no—
216#102—21no—
217C22⋊C8—21no—
218C22⋊C8—31no—
219C22⋊C8—31no—
220C22⋊C8—31no—
221C22⋊C8—31no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8C2×Dic311yescharacteristic
229C22⋊C8C2×Dic311yescharacteristic
230#97—61no—
231C4×C8—61no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—81no—
244C23×C3—81no—
245C23×C3C4×D411yescharacteristic
246C23×C3#12111yescharacteristic
247C23×C3#12311yescharacteristic
248C3×Q8—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—81no—
257C3×D4—81no—
258C3×D4—82no—
259C3×D4—81no—
260C3×D4—81no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—81no—
274C2×C3×C4C2×C22⋊C411yescharacteristic
275C2×C3×C4C4×D411yescharacteristic
276C2×C3×C4#12011yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×Dic3—42no—
279C2×Dic3—81no—
280C2×Dic3—82no—
281C2×Dic3—81no—
282C2×Dic3—82no—
283C3×C8—41no—
284C3×C8—41no—
285C3⋊C8—41no—
286C3⋊C8—41no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C3⋊C8—81no—
290C24—21no—
291C24#23711yescharacteristic
292#41—81no—
293C2×Q8#21711yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—81no—
302C2×D4—81no—
303C2×D4—81no—
304C2×D4#21711yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—62no—
308C22×C4—81no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—241no—
312C22×C4—241no—
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4#23711yescharacteristic
316C22×C4C22×Dic311yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319SD16—81no—
320D8—81no—
321M16—241no—
322M16—241no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—81no—
330C2×C8—121no—
331C2×C8—241no—
332C2×C8—241no—
333C2×C8—241no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4#21711yescharacteristic
339C4⋊C4#21711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—161no—
366C22×C3—161no—
367C22×C3#38611yescharacteristic
368C22×C3#44311yescharacteristic
369C22×C3#44411yescharacteristic
370C22×C3#47111yescharacteristic
371C22×C3#47211yescharacteristic
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—81no—
380Dic3—82no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—81no—
391C23—81no—
392C23D4×Dic311yescharacteristic
393C23#93511yescharacteristic
394C23#93911yescharacteristic
395Q8—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—81no—
404D4—81no—
405D4—82no—
406D4—81no—
407D4—81no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4—122no—
422C2×C4—241no—
423C2×C4—242no—
424C2×C4—241no—
425C2×C4—242no—
426C2×C4D4×Dic311yescharacteristic
427C2×C4#93711yescharacteristic
428C2×C4#93911yescharacteristic
429C2×C4C2×#21711yescharacteristic
430C8—41no—
431C8—41no—
432C8—121no—
433C8—121no—
434C8—121no—
435C8—121no—
436C8—241no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—41no—
442C2×C3—81no—
443C2×C3—161no—
444C2×C3#187611yescharacteristic
445C2×C3#190711yescharacteristic
446C2×C3#191311yescharacteristic
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—161no—
465C22—161no—
466C22#519611yescharacteristic
467C22#519711yescharacteristic
468C22#519911yescharacteristic
469C22#520011yescharacteristic
470C22#522011yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—81no—
479C4—242no—
480C3#926411yescharacteristic, complemented, Sylow
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—161no—
488C2#7184811yescharacteristic
489C2#7208711yescharacteristic
490C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3188C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3188D3
hypocenterC3#9264
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3188

Complements

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

NH ≅ G/Ntype
2. #82920C2semidirect
3. #71708C2semidirect
4. #71676C2semidirect
5. C2×#4537C2semidirect
6. #67680C2semidirect
7. #67674C2semidirect
11. C3×#484C4semidirect
13. C3×#481C4semidirect
16. C3×#450C4semidirect
18. C3×#447C4semidirect
98. #484Dic3semidirect
100. #481Dic3semidirect
103. #450Dic3semidirect
105. #447Dic3semidirect
480. C3#9264semidirect

Maximal subgroups

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

subgroupindex
C3×#31882
#829202
#717082
#716762
C2×#45372
#676802
#676742
#92643