Finite Group Databasebeta

← #10765308

Subgroups — #10765308

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3100C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic, complemented
3#82301C211yesmaximal, characteristic, complemented
4#71801C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#70880C212yesmaximal, complemented
7#70879C212yesmaximal, complemented
8#70878C212yesmaximal, complemented
9D8×C3⋊C8C212yesmaximal, complemented
10#70751C211yesmaximal, characteristic, complemented
11#70743C211yesmaximal, characteristic, complemented
12#17922—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C3×#480C412yes—
16C3×#474C412yes—
17C3×#459C412yes—
18C3×#447C412yes—
19C3×C4×D8C2212yes—
20C3×C4×D8C2212yes—
21C3×#431C2211yescharacteristic
22#5234—22no—
23C2×#830—22no—
24#5012C2212yescomplemented
25#5011—22no—
26D4×C3⋊C8—22no—
27D4×C3⋊C8C2212yes—
28D4×C3⋊C8C2212yes—
29D4×C3⋊C8C2212yescomplemented
30#4926C2211yescharacteristic
31#4925C2211yescharacteristic
32#4925C2211yescharacteristic
33C4×#208C2211yescharacteristic
34D4⋊(C3⋊C8)C2212yes—
35D4⋊(C3⋊C8)C2212yes—
36D4⋊(C3⋊C8)C2212yescomplemented
37D4⋊(C3⋊C8)C2212yescomplemented
38#4458C2212yescomplemented
39#4457C2212yescomplemented
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3100D311yescharacteristic
45#2981—31no—
46#2972—31no—
47#1586—32no—
48#1581—32no—
49#1580—32no—
50C8×D8—32no—
51#1572—31no—
52#1490—31no—
53#1484—31no—
54#1453—31no—
55C2×C3×#41—21no—
56C22×C3×D4C2×C411yescharacteristic
57C23×C3×C4—21no—
58C2×C3×D8C2×C412yes—
59C2×C3×D8C2×C412yes—
60C3×C4⋊Q8C2×C411yescharacteristic
61C3×#127C2×C411yescharacteristic
62C3×#124—21no—
63C3×#121—21no—
64C3×#121C2×C412yes—
65C3×C4×D4—21no—
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4C2×C412yes—
69C3×C4×D4C2312yes—
70C3×C4×D4C2312yes—
71C2×C3×C4⋊C4C2×C411yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#208—21no—
75C2×#208—22no—
76C22×C3⋊C8—22no—
77C22×C3⋊C8—22no—
78C22×C3⋊C8—22no—
79C3×#107C2×C412yes—
80C3×#107C2×C412yes—
81C3×C4⋊C8C2×C412yes—
82C3×#102C2×C412yes—
83C3×#102C2×C412yes—
84C3×#102C2×C412yes—
85C3×#102C2×C412yes—
86C3×C22⋊C8C2×C412yes—
87C3×C4×C8C2311yescharacteristic
88C3×C4×C8C2311yescharacteristic
89#830—21no—
90#830—21no—
91#830—22no—
92#830—22no—
93#830—22no—
94#830—22no—
95#830D412yes—
96C4⋊(C3⋊C8)D412yes—
97C4⋊(C3⋊C8)C2312yes—
98C4⋊(C3⋊C8)C2312yes—
99#803—21no—
100#803—21no—
101C4×C3⋊C8—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#532C2×D311yescharacteristic
107C2×C4×D4C2×D311yescharacteristic
108#480Dic312yes—
109#474Dic312yes—
110#459Dic312yes—
111#447Dic312yes—
112C4×D8C2×D312yes—
113C4×D8C2×D312yes—
114#436—32no—
115#435—62no—
116C8×D4—32no—
117C8×D4—32no—
118C8×D4—32no—
119C8×D4—62no—
120#432—31no—
121#431—31no—
122#431—31no—
123#431C2×D311yescharacteristic
124#408—62no—
125C2×C22⋊C8—62no—
126C4×M16—31no—
127#335—32no—
128#334—32no—
129D4⋊C8—32no—
130D4⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C24×C3—21no—
138C3×#41—82no—
139C2×C3×Q8—21no—
140C2×C3×D4—24no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4C22×C412yes—
146C2×C3×D4C22×C412yes—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—82no—
152C22×C3×C4C22×C411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C3×D8—24no—
156C3×D8—24no—
157C2×C3×C8C22×C412yes—
158C2×C3×C8C22×C412yes—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4C22×C412yes—
162C3×C4⋊C4C22×C412yes—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—22no—
166C3×C22⋊C4—22no—
167C3×C42C22×C411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42#4111yescharacteristic
170C3×C42C2411yescharacteristic
171#208—42no—
172#208—82no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—82no—
178C2×C3⋊C8—82no—
179C2×C3⋊C8—82no—
180C2×C3⋊C8C2×D412yes—
181C2×C3⋊C8C2×D412yes—
182C2×#41—21no—
183C22×D4C2×Dic311yescharacteristic
184C23×C4—21no—
185C2×D8C2×Dic312yes—
186C2×D8C2×Dic312yes—
187C2×M16—62no—
188C2×M16—61no—
189C22×C8—62no—
190C22×C8—62no—
191C22×C8—62no—
192C4⋊Q8C2×Dic311yescharacteristic
193#127C2×Dic311yescharacteristic
194#124—21no—
195#121—21no—
196#121C2×Dic312yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C2×Dic312yes—
201C4×D4C22×D312yes—
202C4×D4C22×D312yes—
203C2×C4⋊C4C2×Dic311yescharacteristic
204C2×C22⋊C4—21no—
205C2×C42C22×D311yescharacteristic
206#107C2×Dic312yes—
207#107C2×Dic312yes—
208C4⋊C8—32no—
209C4⋊C8—32no—
210C4⋊C8—32no—
211C4⋊C8C2×Dic312yes—
212#102C2×Dic312yes—
213#102C2×Dic312yes—
214#102C2×Dic312yes—
215#102C2×Dic312yes—
216C22⋊C8—32no—
217C22⋊C8—62no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8C2×Dic312yes—
224#97—61no—
225#97—61no—
226C4×C8—31no—
227C4×C8—31no—
228C4×C8—31no—
229C4×C8—31no—
230C4×C8—61no—
231C4×C8C22×D311yescharacteristic
232C4×C8C22×D311yescharacteristic
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—81no—
237C23×C3—81no—
238C23×C3C4×D411yescharacteristic
239C3×Q8—81no—
240C3×D4—24no—
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—82no—
247C3×D4#13114yes—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—82no—
258C2×C3×C4—81no—
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4C23×C411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C3×C8—22no—
266C3×C8—22no—
267C3⋊C8—22no—
268C3⋊C8—22no—
269C3⋊C8—22no—
270C3⋊C8—41no—
271C3⋊C8—42no—
272C3⋊C8—42no—
273C24—21no—
274#41—82no—
275C2×Q8—21no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4C22×Dic312yes—
282C2×D4C22×Dic312yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—82no—
288C22×C4D3×D411yescharacteristic
289C22×C4#23711yescharacteristic
290C22×C4C22×Dic311yescharacteristic
291D8—24no—
292D8—24no—
293M16—122no—
294M16—242no—
295C2×C8—32no—
296C2×C8—32no—
297C2×C8—61no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—61no—
301C2×C8—242no—
302C2×C8—242no—
303C2×C8—242no—
304C2×C8C22×Dic312yes—
305C2×C8C22×Dic312yes—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4C22×Dic312yes—
309C4⋊C4C22×Dic312yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C42D3×D411yescharacteristic
315C42#23711yescharacteristic
316C42C22×Dic311yescharacteristic
317C42C23×D311yescharacteristic
318C22×C3—21no—
319C22×C3—24no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—84no—
328C22×C3C2×C4×D411yescharacteristic
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4C2×#13112yes—
336C3×C4#57612yes—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—81no—
341C23—81no—
342C23D4×Dic311yescharacteristic
343Q8—81no—
344D4—24no—
345D4—22no—
346D4—22no—
347D4—22no—
348D4—81no—
349D4—81no—
350D4—82no—
351D4#94814yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4D4×Dic311yescharacteristic
364C2×C4D4×Dic311yescharacteristic
365C2×C4D4×Dic311yescharacteristic
366C2×C4C2×D3×D411yescharacteristic
367C2×C4C2×#23711yescharacteristic
368C2×C4C23×Dic311yescharacteristic
369C8—22no—
370C8—22no—
371C8—62no—
372C8—62no—
373C8—62no—
374C8—121no—
375C8—122no—
376C8—122no—
377C2×C3—41no—
378C2×C3—44no—
379C2×C3—82no—
380C2×C3#292911yescharacteristic
381C2×C3#294811yescharacteristic
382C2×C3#295611yescharacteristic
383C22—21no—
384C22—21no—
385C22—21no—
386C22—24no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22C2×D4×Dic311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—42no—
400C4#575712yes—
401C4C2×#94812yes—
402C3#1792211yescharacteristic, complemented, Sylow
403C2—41no—
404C2—44no—
405C2—82no—
406C2#8291611yescharacteristic
407C2#8382311yescharacteristic
408C2#8400011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3C2×C4×D4
cosocleC3×C42C24
hypercenter#3100D3
hypocenterC3#17922
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. #82301C2semidirect
4. #71801C2semidirect
5. #71599C2semidirect
6. #70880C2semidirect
7. #70879C2semidirect
8. #70878C2semidirect
9. D8×C3⋊C8C2semidirect
10. #70751C2semidirect
11. #70743C2semidirect
24. #5012C22semidirect
29. D4×C3⋊C8C22semidirect
36. D4⋊(C3⋊C8)C22semidirect
37. D4⋊(C3⋊C8)C22semidirect
38. #4458C22semidirect
39. #4457C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
402. C3#17922semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#832802
#823012
#718012
#715992
#708802
#708792
#708782
D8×C3⋊C82
#707512
#707432
#179223