Finite Group Databasebeta

← #10697544

Subgroups — #10697544

Order 768.

794 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#3036C211yesmaximal, characteristic, complemented
2#83302C211yesmaximal, characteristic, complemented
3#82378C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#70502C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10238—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#940—21no—
13C2×#930—22no—
14C2×#874C2211yescharacteristic
15C3×#463—22no—
16C3×#461—21no—
17C3×#459—21no—
18C3×#423C2211yescharacteristic
19C3×#417—21no—
20#5088—21no—
21#5073—21no—
22#5036—22no—
23#5015—22no—
24#4988—21no—
25#4960—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30D4⋊(C3⋊C8)—22no—
31#4485—22no—
32#3205—31no—
33#3036D311yescharacteristic, complemented
34#2541—31no—
35#1545—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×#67—24no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12D412yes—
46C3×AGL(1,ℤ/8)—42no—
47C3×AGL(1,ℤ/8)—42no—
48C2×C3×SD16—22no—
49C2×C3×D8—22no—
50C2×C3×M16—21no—
51C3×C4⋊Q8D411yescharacteristic, complemented
52C3×#127D411yescharacteristic, complemented
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C42C2311yescharacteristic
65#940—42no—
66#940—42no—
67C2×#213—24no—
68#930—48no—
69C2×(C2×D3)⋊C4—24no—
70C2×C4⋊Dic3—21no—
71C2×C4×Dic3—21no—
72#874—21no—
73#874—21no—
74#874—21no—
75#874—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yescomplemented
78C3×#103—22no—
79C3×#102—22no—
80C3⋊#95D412yes—
81#830—21no—
82#830—21no—
83#809—24no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yescomplemented
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C2×AGL(1,ℤ/8)—21no—
89#532C2×D311yescharacteristic
90C2×#127—31no—
91C2×#127—61no—
92C2×#121—62no—
93C2×C4×D4C2×D311yescharacteristic
94#493—61no—
95#486—61no—
96#463—22no—
97#461—21no—
98#459—21no—
99#459—62no—
100#447—62no—
101#431—31no—
102#431—31no—
103#423C2×D311yescharacteristic
104#417—21no—
105#392—62no—
106#390—61no—
107C4×C4⋊C4—31no—
108#331—62no—
109#327—62no—
110D4⋊C8—62no—
111C24×C3—21no—
112C23×D3—22no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C2×C3×D4D812yescomplemented
128C22×C3×C4—21no—
129C22×C3×C4—44no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C2×#67—48no—
137C2×#67—48no—
138C2×#67—84no—
139C22×Dic3—22no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—44no—
143C2×D12—41no—
144C2×D12—41no—
145C2×D12—41no—
146C2×D12—41no—
147C2×D12—41no—
148C2×D12—41no—
149C2×D12—41no—
150C2×D12—41no—
151C2×D12—82no—
152C3×SD16—44no—
153C3×D8—44no—
154C3×M16—41no—
155C3×M16—41no—
156C2×C3×C8—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169#213—84no—
170(C2×D3)⋊C4—48no—
171C4⋊Dic3—24no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C2×#41—21no—
177C22×D4—32no—
178C22×D4—64no—
179C22×D4—61no—
180C22×D4—61no—
181C22×D4#6711yescharacteristic
182C23×C4—21no—
183AGL(1,ℤ/8)—42no—
184AGL(1,ℤ/8)—42no—
185C2×SD16—22no—
186C2×D8—22no—
187C2×D8—64no—
188C2×M16—21no—
189C4⋊Q8#6711yescharacteristic, complemented
190#127—61no—
191#127—61no—
192#127—61no—
193#127—61no—
194#127—122no—
195#127—122no—
196#127#6711yescharacteristic, complemented
197#124—21no—
198#121—21no—
199#121—21no—
200#121—128no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—64no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8#6712yescomplemented
217#103—22no—
218#102—22no—
219#102—64no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C4×C8—61no—
223C4×C8—61no—
224#95—32no—
225C23×C3—22no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3#12011yescharacteristic
232C22×D3—44no—
233C22×D3—44no—
234C22×D3—42no—
235C22×D3—42no—
236C22×D3—42no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—42no—
245C3×D4—41no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—81no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—82no—
255C2×C3×C4—82no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12711yescharacteristic
264C2×C3×C4C2×D811yescharacteristic
265C2×C3×C4#13511yescharacteristic
266C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268#67—816no—
269C2×Dic3—24no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272D12—44no—
273D12—82no—
274D12—82no—
275D12—82no—
276D12—81no—
277D12—81no—
278D12—81no—
279D12—81no—
280C3×C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284C24—62no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289C2×Q8—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—62no—
300C2×D4—62no—
301C2×D4—124no—
302C2×D4—128no—
303C2×D4—128no—
304C2×D4—121no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—121no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—121no—
312C2×D4—242no—
313C2×D4—244no—
314C2×D4#21312yescomplemented
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—62no—
321C22×C4D3×D411yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4C2×#6711yescharacteristic
324SD16—44no—
325D8—44no—
326D8—244no—
327M16—41no—
328M16—41no—
329C2×C8—22no—
330C2×C8—62no—
331C2×C8—121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—64no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—128no—
342C42—121no—
343C42—121no—
344C42D3×D411yescharacteristic
345C42#23711yescharacteristic
346C42C2×#6711yescharacteristic
347C42C2×#6711yescharacteristic
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3#39211yescharacteristic
359C22×C3#44711yescharacteristic
360C22×C3#44911yescharacteristic
361C2×D3—84no—
362C2×D3—84no—
363C2×D3—82no—
364C2×D3—82no—
365C2×D3—82no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—81no—
370C3×C4—81no—
371C3×C4#44711yescharacteristic
372C3×C4#44911yescharacteristic
373C3×C4#45312yes—
374C3×C4#46611yescharacteristic
375C3×C4#46911yescharacteristic
376Dic3—44no—
377C23—22no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—124no—
384C23—124no—
385C23—122no—
386C23—122no—
387C23—122no—
388C23#93711yescharacteristic
389Q8—41no—
390Q8—41no—
391D4—22no—
392D4—22no—
393D4—42no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—124no—
402D4—242no—
403D4—242no—
404D4—242no—
405D4—2416no—
406D4—241no—
407D4—241no—
408D4—241no—
409D4—241no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—64no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4C2×#21311yescharacteristic
423C2×C4#93211yescharacteristic
424C2×C4#93711yescharacteristic
425C2×C4#93811yescharacteristic
426C2×C4#93911yescharacteristic
427C2×C4#93911yescharacteristic
428C2×C4#94011yescharacteristic
429C2×C4#94911yescharacteristic
430C2×C4#95011yescharacteristic
431C2×C4#95311yescharacteristic
432C8—42no—
433C8—122no—
434C8—241no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—82no—
438C2×C3—161no—
439C2×C3#199911yescharacteristic
440C2×C3#200211yescharacteristic
441C2×C3#200811yescharacteristic
442D3—162no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—42no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—244no—
454C22—244no—
455C22—242no—
456C22—242no—
457C22—242no—
458C22#503611yescharacteristic
459C22#503711yescharacteristic
460C22#522311yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—81no—
465C4—81no—
466C4—124no—
467C4#503811yescharacteristic
468C4#503911yescharacteristic
469C4#519812yes—
470C4#521711yescharacteristic
471C4#523811yescharacteristic
472C3#1023811yescharacteristic, complemented, Sylow
473C2—21no—
474C2—21no—
475C2—82no—
476C2—161no—
477C2—482no—
478C2#7168411yescharacteristic
479C2#7209411yescharacteristic
480C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10238
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C3×#3036C2semidirect
2. #83302C2semidirect
3. #82378C2semidirect
4. #71811C2semidirect
5. #71597C2semidirect
6. #70502C2semidirect
7. #67715C2semidirect
33. #3036D3semidirect
51. C3×C4⋊Q8D4semidirect
52. C3×#127D4semidirect
77. C3×C4⋊C8D4semidirect
85. C4⋊(C3⋊C8)D4semidirect
127. C2×C3×D4D8semidirect
189. C4⋊Q8#67semidirect
196. #127#67semidirect
216. C4⋊C8#67semidirect
314. C2×D4#213semidirect
472. C3#10238semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833022
#823782
#718112
#715972
#705022
#677152
#102383