Finite Group Databasebeta

← #10696029

Subgroups — #10696029

Order 768.

734 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82347C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic, complemented
3#79297C211yesmaximal, characteristic, complemented
4C3×#1486C211yesmaximal, characteristic, complemented
5#70513C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10012—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×C4×#67—22no—
11#5477C2211yescharacteristic
12C2×C4×D12C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5223—21no—
16#5217—21no—
17C2×#825—22no—
18#5015—21no—
19C4×#213—22no—
20#5007—21no—
21#4993—21no—
22#4967—21no—
23#4960—21no—
24#4949—21no—
25#4932C2211yescharacteristic
26#4925C2211yescharacteristic
27#4846—21no—
28#4843—21no—
29#4736—21no—
30#4732—21no—
31C4×D24—22no—
32C3×D4⋊C8—22no—
33#4483—22no—
34#3100—31no—
35#3100—31no—
36#2397—31no—
37#1963—31no—
38#1486—31no—
39#1486D311yescharacteristic, complemented
40#1274—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C2×#235—21no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C2×C3×M16—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#217—22no—
56C2×#213—22no—
57#930—21no—
58#930—21no—
59C4×#67—44no—
60C4×#67—44no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3D412yes—
66C2×D24—22no—
67#875—21no—
68#874D411yescharacteristic, complemented
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic, complemented
76C3×C4⋊C8—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C4×C8D412yescomplemented
81#832—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#825—44no—
86#821—22no—
87#821—22no—
88#818—22no—
89#809—22no—
90#807—22no—
91C4⋊(C3⋊C8)—21no—
92C4×C3⋊C8D412yescomplemented
93#532—31no—
94C2×C4×D4—31no—
95C2×C4×D4—62no—
96C2×C4×D4C2×D311yescharacteristic
97#480—61no—
98#480—61no—
99#474—61no—
100#474—61no—
101#459—61no—
102#459—61no—
103#447—61no—
104#447—61no—
105C4×D8—62no—
106C4×D8—62no—
107#431—31no—
108#431C2×D311yescharacteristic
109#423C2×D311yescharacteristic
110C2×#104—62no—
111#399—61no—
112#399—61no—
113#392—61no—
114#392—61no—
115#376—31no—
116D4⋊C8—22no—
117D4⋊C8—62no—
118C24×C3—21no—
119C23×D3—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—42no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C2×#67—44no—
132C2×#67—44no—
133C2×#67—44no—
134C2×#67—41no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137#235—42no—
138#235—42no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C2×C4×D3—42no—
149C2×C4×D3—42no—
150C2×Dic6—21no—
151C3×M16—42no—
152C2×C3×C8—22no—
153C2×C3×C8—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162#217—44no—
163#213—44no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173Dic3⋊C4—44no—
174C4×Dic3—22no—
175C4×Dic3D814yescomplemented
176C2×C3⋊C8—22no—
177D24—44no—
178C2×#41—61no—
179C22×D4—31no—
180C22×D4—62no—
181C22×D4D1211yescharacteristic
182C23×C4—21no—
183C23×C4—61no—
184C2×D8—62no—
185C2×D8—62no—
186C2×M16—21no—
187C4⋊Q8—31no—
188#127—31no—
189#124—61no—
190#121—61no—
191#121—61no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—61no—
197C4×D4—61no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—124no—
203C4×D4—124no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4D1211yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C42—32no—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214#107—62no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—61no—
219#104—124no—
220#102—62no—
221#102—62no—
222#102—62no—
223#102—62no—
224C22⋊C8—21no—
225C22⋊C8—61no—
226C4×C8—32no—
227C4×C8#6712yescomplemented
228#95—61no—
229#95—61no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—42no—
235C23×C3#12011yescharacteristic
236C22×D3—22no—
237C22×D3—41no—
238C22×D3—42no—
239C22×D3—41no—
240C22×D3—42no—
241C22×D3—41no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—41no—
245C3×D4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—82no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12711yescharacteristic
259C2×C3×C4C2×D812yes—
260C2×C3×C4AGL(1,ℤ/8)12yes—
261#67—416no—
262#67—82no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269D12—22no—
270D12—22no—
271D12—41no—
272D12—41no—
273D12—42no—
274D12—42no—
275D12—42no—
276C4×D3—82no—
277C4×D3—81no—
278C4×D3—81no—
279C4×D3—81no—
280C4×D3—81no—
281Dic6—42no—
282C3×C8—42no—
283C3×C8—81no—
284C3⋊C8—42no—
285C24—21no—
286C24—61no—
287#41—122no—
288#41—122no—
289C2×Q8—61no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—122no—
299C2×D4—124no—
300C2×D4—124no—
301C2×D4—124no—
302C2×D4—121no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—62no—
309C22×C4—61no—
310C22×C4—121no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4C2×D1211yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318D8—124no—
319D8—124no—
320M16—42no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—62no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—124no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—124no—
334C22⋊C4—124no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C42—34no—
340C42—62no—
341C42#23511yescharacteristic
342C42D3×D411yescharacteristic
343C42D3×D411yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3#39211yescharacteristic
353C22×C3#44712yes—
354C2×D3—44no—
355C2×D3—42no—
356C2×D3—82no—
357C2×D3—81no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—81no—
367C3×C4#45912yes—
368C3×C4#49311yescharacteristic
369C3×C4#49611yescharacteristic
370Dic3—44no—
371Dic3—81no—
372Dic3—81no—
373C23—22no—
374C23—41no—
375C23—42no—
376C23—41no—
377C23—42no—
378C23—62no—
379C23—121no—
380C23—122no—
381C23—121no—
382C23—122no—
383C23—121no—
384C23#88211yescharacteristic
385Q8—122no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390D4—62no—
391D4—62no—
392D4—121no—
393D4—121no—
394D4—122no—
395D4—122no—
396D4—1216no—
397D4—122no—
398D4—242no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—64no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—122no—
412C2×C4—241no—
413C2×C4—241no—
414C2×C4—241no—
415C2×C4—241no—
416C2×C4—242no—
417C2×C4#88211yescharacteristic
418C2×C4#88411yescharacteristic
419C2×C4#88411yescharacteristic
420C2×C4D3×D812yes—
421C2×C4#91412yes—
422C2×C4#93011yescharacteristic
423C2×C4#93711yescharacteristic
424C2×C4#94011yescharacteristic
425C8—42no—
426C8—81no—
427C8—122no—
428C2×C3—22no—
429C2×C3—82no—
430C2×C3#201011yescharacteristic
431C2×C3#201011yescharacteristic
432C2×C3#201111yescharacteristic
433D3—82no—
434D3—161no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22—42no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—124no—
443C22—122no—
444C22—242no—
445C22—241no—
446C22—241no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22#477312yes—
451C22#496011yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—81no—
457C4—124no—
458C4—241no—
459C4—241no—
460C4#478612yes—
461C4#515111yescharacteristic
462C4#517611yescharacteristic
463C3#1001211yescharacteristic, complemented, Sylow
464C2—22no—
465C2—82no—
466C2—242no—
467C2—481no—
468C2#6999311yescharacteristic
469C2#7020911yescharacteristic
470C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#392
hypercenter#1486D3
hypocenterC3#10012
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #82347C2semidirect
2. #82308C2semidirect
3. #79297C2semidirect
4. C3×#1486C2semidirect
5. #70513C2semidirect
6. #70496C2semidirect
7. #67737C2semidirect
39. #1486D3semidirect
68. #874D4semidirect
75. #869D4semidirect
80. C3×C4×C8D4semidirect
92. C4×C3⋊C8D4semidirect
175. C4×Dic3D8semidirect
227. C4×C8#67semidirect
463. C3#10012semidirect

Maximal subgroups

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

subgroupindex
#823472
#823082
#792972
C3×#14862
#705132
#704962
#677372
#100123