Finite Group Databasebeta

← #10777926

Subgroups — #10777926

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81775C211yesmaximal, characteristic, complemented
2#80313C212yesmaximal, complemented
3#79517C211yesmaximal, characteristic, complemented
4C3×#1687C211yesmaximal, characteristic, complemented
5#71287C211yesmaximal, characteristic, complemented
6#71010C212yesmaximal, complemented
7#70874C211yesmaximal, characteristic, complemented
8#68553C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68414C212yesmaximal, complemented
11#68382C212yesmaximal, complemented
12#19160—31nomaximal, Sylow
13#5716C2212yes—
14C4×#237C2211yescharacteristic
15C3×#493C2211yescharacteristic
16C3×#459C2212yes—
17C3×#436C2211yescharacteristic
18#5158—22no—
19#5157—21no—
20#5157—21no—
21Dic3×D8—22no—
22Dic3×D8—21no—
23Dic3×D8—21no—
24#5127—21no—
25Dic3×M16—21no—
26#5075C2212yes—
27#5070C2212yes—
28#5012C2211yescharacteristic
29#5008C2212yes—
30#4850C2212yes—
31#4833C2211yescharacteristic
32#4833C2212yes—
33#4831C2211yescharacteristic
34#4784—22no—
35#4758—22no—
36#4728—21no—
37#4718—21no—
38C4×D3×C8C2211yescharacteristic
39#4682C2211yescharacteristic
40#4679C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#331C2212yes—
43#4544C2212yes—
44#4481C2212yes—
45#4481C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3169—31no—
49#3106—32no—
50#2965—31no—
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695—31no—
55#1688—31no—
56#1687D311yescharacteristic, complemented
57#1564—31no—
58#1543—32no—
59C2×#237—22no—
60C2×#237D411yescharacteristic
61C2×C3×D8—22no—
62C2×C3×D8—21no—
63C2×C3×D8—21no—
64C2×C3×M16—21no—
65C3×#127C2312yes—
66C3×C4×D4C2311yescharacteristic
67#938—22no—
68#938—22no—
69#936—22no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72D4×Dic3D411yescharacteristic
73C2×#214—22no—
74C4×#67—21no—
75C2×C4×Dic3—21no—
76C2×#208—21no—
77C2×D3×C8D411yescharacteristic
78C2×D3×C8D411yescharacteristic
79#892D411yescharacteristic
80#887D411yescharacteristic
81#67⋊C4—21no—
82#877—21no—
83C42×D3C2311yescharacteristic
84#869C2312yes—
85C4×Dic6C2311yescharacteristic
86C3×C4⋊C8C2312yes—
87C3×C4⋊C8C2311yescharacteristic
88C3×#102—22no—
89C3×C22⋊C8—21no—
90C3×C4×C8C2311yescharacteristic
91#832—22no—
92#832—22no—
93#832—22no—
94#832—22no—
95#830—21no—
96(C2×D3)⋊C8—21no—
97(C2×D3)⋊C8—22no—
98#818—21no—
99#818—21no—
100#818—22no—
101#816—22no—
102#815—21no—
103#815—21no—
104Dic3⋊C8—21no—
105Dic3⋊C8—21no—
106C8×Dic3—22no—
107C8×Dic3—21no—
108C8×Dic3D411yescharacteristic
109C8×Dic3D411yescharacteristic
110#808—22no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8C2311yescharacteristic
114#532—32no—
115C4×#41—31no—
116#500—31no—
117#493C2×D311yescharacteristic
118#486—32no—
119#482—62no—
120#477—32no—
121#476—32no—
122#466—61no—
123#466—61no—
124#460—32no—
125#459C2×D312yes—
126#449—62no—
127#446—31no—
128#446—31no—
129C4×SD16—62no—
130C4×D8—61no—
131C4×D8—61no—
132C4×D8—62no—
133#436—31no—
134#436C2×D311yescharacteristic
135#435—61no—
136#435—61no—
137#431—31no—
138#431—32no—
139#431—61no—
140C4×M16—61no—
141C2×C4×C8—31no—
142#335—32no—
143#334—31no—
144#334C2×D311yescharacteristic
145#333—32no—
146#333—32no—
147#331—32no—
148#331C2×D312yes—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4C2×D411yescharacteristic
153C22×C3×C4—21no—
154C2×#67—21no—
155C2×#67—42no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—42no—
159#237—42no—
160#237—84no—
161C2×C4×D3C2×D411yescharacteristic
162C2×C4×D3C2×D411yescharacteristic
163C2×C4×D3#4111yescharacteristic
164C2×Dic6—22no—
165C2×Dic6C2×D411yescharacteristic
166C3×D8—42no—
167C3×D8—44no—
168C3×D8—42no—
169C3×M16—42no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D411yescharacteristic
174C3×C4⋊C4C2×D411yescharacteristic
175C3×C22⋊C4—21no—
176C3×C42C2411yescharacteristic
177#217—21no—
178#217—42no—
179#217—42no—
180#214—44no—
181(C2×D3)⋊C4—21no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3C2×D411yescharacteristic
185Dic3⋊C4—21no—
186Dic3⋊C4—21no—
187C4×Dic3—21no—
188C4×Dic3—21no—
189C4×Dic3—21no—
190C4×Dic3C2×D411yescharacteristic
191C4×Dic3C2×D411yescharacteristic
192C4×Dic3#4111yescharacteristic
193#208—42no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8C2×D411yescharacteristic
197C2×C3⋊C8C2×D411yescharacteristic
198D3×C8—22no—
199D3×C8—41no—
200C2×#41—31no—
201C2×#41—62no—
202C2×SD16—62no—
203C2×D8—22no—
204C2×D8—21no—
205C2×D8—21no—
206C2×M16—21no—
207C2×M16—61no—
208C22×C8—31no—
209C22×C8—31no—
210C4⋊Q8—32no—
211#127C22×D312yes—
212#124—62no—
213#121—62no—
214#121—62no—
215C4×Q8—31no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—62no—
219C4×D4—62no—
220C4×D4—61no—
221C4×D4—61no—
222C4×D4C22×D311yescharacteristic
223#117—31no—
224#117—61no—
225C2×C42—31no—
226C2×C42—61no—
227#107—62no—
228#107—61no—
229#107—61no—
230#106—62no—
231C4⋊C8—31no—
232C4⋊C8—32no—
233C4⋊C8—61no—
234C4⋊C8—61no—
235C4⋊C8C22×D311yescharacteristic
236C4⋊C8C22×D312yes—
237#103—62no—
238#102—22no—
239#102—62no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—21no—
244C22⋊C8—61no—
245C22⋊C8—62no—
246C22⋊C8—61no—
247#97—61no—
248#97—61no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—31no—
252C4×C8—61no—
253C4×C8—62no—
254C4×C8C22×D311yescharacteristic
255C23×C3—21no—
256C23×C3—42no—
257C22×D3#12111yescharacteristic
258C3×D4—22no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—82no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270#67—42no—
271#67—84no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—84no—
278C2×Dic3#12111yescharacteristic
279C2×Dic3#12111yescharacteristic
280C2×Dic3#12111yescharacteristic
281C4×D3—21no—
282C4×D3—21no—
283C4×D3—22no—
284C4×D3—22no—
285Dic6—22no—
286Dic6—82no—
287C3×C8—21no—
288C3×C8—41no—
289C3×C8—42no—
290C3×C8#13512yes—
291C3⋊C8—21no—
292C3⋊C8—21no—
293C3⋊C8—41no—
294C3⋊C8—42no—
295#41—122no—
296#41—244no—
297C2×Q8—31no—
298C2×Q8—62no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—61no—
303C2×D4—122no—
304C2×D4D3×D411yescharacteristic
305C22×C4—21no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—122no—
312SD16—124no—
313D8—42no—
314D8—44no—
315D8—42no—
316M16—42no—
317M16—122no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—31no—
321C2×C8—31no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8D3×D411yescharacteristic
327C2×C8D3×D411yescharacteristic
328C4⋊C4—31no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4D3×D411yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—61no—
343C42—61no—
344C42—61no—
345C42C23×D311yescharacteristic
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3C2×#12111yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C3×C4—21no—
357C3×C4—41no—
358C3×C4C2×#13511yescharacteristic
359C3×C4#57511yescharacteristic
360C3×C4#57512yes—
361Dic3—21no—
362Dic3—21no—
363Dic3—41no—
364Dic3—41no—
365Dic3—82no—
366Dic3#47212yes—
367C23—21no—
368C23—31no—
369C23—42no—
370Q8—62no—
371Q8—242no—
372D4—22no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—82no—
378D4—122no—
379D4—244no—
380C2×C4—21no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—62no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—244no—
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×D3×D411yescharacteristic
398C2×C4D3×#4111yescharacteristic
399C8—21no—
400C8—41no—
401C8—42no—
402C8—61no—
403C8—61no—
404C8—121no—
405C8—122no—
406C8#91212yes—
407C2×C3—81no—
408C2×C3—162no—
409C2×C3#303511yescharacteristic
410C2×C3#305411yescharacteristic
411C2×C3C2×#47211yescharacteristic
412D3—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22D3×#12111yescharacteristic
423C4—21no—
424C4—32no—
425C4—41no—
426C4—61no—
427C4—61no—
428C4—121no—
429C4—121no—
430C4—242no—
431C4C2×#91211yescharacteristic
432C4#577311yescharacteristic
433C4#577312yes—
434C3#1916011yescharacteristic, complemented, Sylow
435C2—81no—
436C2—121no—
437C2—162no—
438C2#8033111yescharacteristic
439C2#8086311yescharacteristic
440C2D3×#47211yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1687C2
socleC22×C3C2×#121
hypercenter#1687D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1687

Complements

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

NH ≅ G/Ntype
1. #81775C2semidirect
2. #80313C2semidirect
3. #79517C2semidirect
4. C3×#1687C2semidirect
5. #71287C2semidirect
6. #71010C2semidirect
7. #70874C2semidirect
8. #68553C2semidirect
9. #68527C2semidirect
10. #68414C2semidirect
11. #68382C2semidirect
56. #1687D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#817752
#803132
#795172
C3×#16872
#712872
#710102
#708742
#685532
#685272
#684142
#683822
#191603