Finite Group Databasebeta

← #10695972

Subgroups — #10695972

Order 768.

692 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 83 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#83306C211yesmaximal, characteristic, complemented
2#83300C211yesmaximal, characteristic, complemented
3#73798C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10004—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#897—21no—
11C2×#895—21no—
12C2×#875C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C2×C3×#104—22no—
15C3×C4×M16C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17#5089—21no—
18#5084—21no—
19#5074—21no—
20#5072—21no—
21#5040—21no—
22#5036—21no—
23#5036—21no—
24#5034—21no—
25#5016—22no—
26#5014—22no—
27#4939—22no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4677—21no—
31#4671—21no—
32#4464—22no—
33#4459—22no—
34#3213—31no—
35#3109—31no—
36#2540—31no—
37#1980D311yescharacteristic, complemented
38#1527—31no—
39#1490—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×D12—22no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×Dic6D411yescharacteristic
47C2×C3×M16—21no—
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#127D411yescharacteristic, complemented
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42D412yes—
57C2×C3×C42C2311yescharacteristic
58C2×#214—22no—
59C2×#213—22no—
60C2×#213—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3D411yescharacteristic
68#897—44no—
69#895—44no—
70#875—21no—
71#875—21no—
72#875—21no—
73#875—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12—21no—
78C3×#104—44no—
79C3×#97—21no—
80C3×C4×C8—21no—
81C3×#95—21no—
82#832—22no—
83#830—21no—
84#830—21no—
85#810—22no—
86#809—22no—
87#809—22no—
88#808—22no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yescomplemented
91#803—21no—
92#803D412yescomplemented
93#532C2×D311yescharacteristic
94C2×#124—31no—
95C2×C22⋊Q8—61no—
96C2×#121—61no—
97C2×C4×D4—31no—
98#496—61no—
99#489—61no—
100#481—61no—
101#476—61no—
102#463—62no—
103#459—61no—
104#448—61no—
105#447—61no—
106#447—61no—
107#442—62no—
108#432—31no—
109#432—31no—
110C2×#104—22no—
111C4×M16C2×D311yescharacteristic
112#399—61no—
113#392—61no—
114#390—62no—
115C4×C4⋊C4C2×D311yescharacteristic
116#329—62no—
117D4⋊C8—62no—
118C23×D3—21no—
119C23×D3—21no—
120C3×#41—42no—
121C3×#41—42no—
122C2×C3×Q8—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C2×D12—42no—
140C2×D12—44no—
141C2×D12—44no—
142C2×D12—82no—
143C2×C4×D3—42no—
144C2×C4×D3—42no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×M16—22no—
151C3×M16—41no—
152C2×C3×C8—41no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42—42no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#214—82no—
164#213—44no—
165#213—82no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—44no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C2×#41—21no—
180C22×Q8—31no—
181C22×D4—31no—
182C22×D4—31no—
183C22×D4—62no—
184C23×C4—61no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×D8—62no—
188C2×M16—21no—
189C4⋊Q8D1211yescharacteristic, complemented
190#127D1211yescharacteristic, complemented
191#124—21no—
192#124—61no—
193#124—61no—
194#124—61no—
195#124—61no—
196C22⋊Q8—124no—
197#121—21no—
198#121—21no—
199#121—124no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—61no—
203C4×D4—61no—
204C4×D4—61no—
205C4×D4—61no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C42#6712yes—
215C2×C42C22×D311yescharacteristic
216#106—62no—
217C4⋊C8—32no—
218C4⋊C8—61no—
219#104—44no—
220#103—62no—
221#102—62no—
222#102—62no—
223#102—62no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226#97—21no—
227#97—32no—
228#97—61no—
229C4×C8—21no—
230#95—21no—
231C23×C3—41no—
232C23×C3#12011yescharacteristic
233C22×D3—22no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C3×Q8—42no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—81no—
250C3×D4—81no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—82no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12711yescharacteristic
266C2×Dic3—41no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271D12—22no—
272D12—22no—
273D12—41no—
274D12—41no—
275D12—42no—
276D12—81no—
277D12—88no—
278C4×D3—82no—
279C4×D3—82no—
280Dic6—42no—
281Dic6—81no—
282C3×C8—42no—
283C3⋊C8—42no—
284C3⋊C8—42no—
285C24—61no—
286C24—61no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—61no—
291C2×Q8—61no—
292C2×Q8—122no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—122no—
302C2×D4—124no—
303C2×D4—122no—
304C2×D4—124no—
305C2×D4—242no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—242no—
318D8—124no—
319D8—242no—
320M16—22no—
321M16—41no—
322C2×C8—41no—
323C2×C8—62no—
324C2×C8—62no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
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—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C42—42no—
342C42C2×D1211yescharacteristic
343C42#23511yescharacteristic
344C42D3×D411yescharacteristic
345C42D3×D411yescharacteristic
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#39211yescharacteristic
352C22×C3#45311yescharacteristic
353C22×C3#45411yescharacteristic
354C2×D3—42no—
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—81no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—44no—
372C3×C4—81no—
373C3×C4#45312yes—
374Dic3—81no—
375Dic3—81no—
376C23—41no—
377C23—62no—
378C23—121no—
379C23—121no—
380C23—121no—
381C23—122no—
382C23—122no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23#93711yescharacteristic
390Q8—42no—
391Q8—122no—
392Q8—241no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—62no—
397D4—62no—
398D4—81no—
399D4—81no—
400D4—121no—
401D4—121no—
402D4—122no—
403D4—241no—
404D4—248no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—242no—
420C2×C4—242no—
421C2×C4#88211yescharacteristic
422C2×C4#88211yescharacteristic
423C2×C4#88411yescharacteristic
424C2×C4#88411yescharacteristic
425C2×C4#93011yescharacteristic
426C2×C4#94011yescharacteristic
427C8—42no—
428C8—122no—
429C8—122no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—161no—
433C2×C3#200211yescharacteristic
434C2×C3#200611yescharacteristic
435C2×C3#201011yescharacteristic
436D3—82no—
437D3—161no—
438C22—21no—
439C22—21no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22—241no—
447C22—241no—
448C22—242no—
449C22—242no—
450C22—241no—
451C22—241no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22#496011yescharacteristic
456C22#506211yescharacteristic
457C22#507711yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—81no—
465C4—241no—
466C4—241no—
467C4#482212yes—
468C3#1000411yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—161no—
472C2—242no—
473C2—481no—
474C2#6966411yescharacteristic
475C2#7038511yescharacteristic
476C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#392
hypercenter#1980D3
hypocenterC3#10004
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
1. #83306C2semidirect
2. #83300C2semidirect
3. #73798C2semidirect
4. C3×#1980C2semidirect
5. #69860C2semidirect
6. #69855C2semidirect
7. #67621C2semidirect
37. #1980D3semidirect
48. C3×C4⋊Q8D4semidirect
49. C3×#127D4semidirect
90. C4⋊(C3⋊C8)D4semidirect
92. #803D4semidirect
189. C4⋊Q8D12semidirect
190. #127D12semidirect
468. C3#10004semidirect

Maximal subgroups

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

subgroupindex
#833062
#833002
#737982
C3×#19802
#698602
#698552
#676212
#100043