Finite Group Databasebeta

← #10697548

Subgroups — #10697548

Order 768.

794 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 103 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#82378C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic, complemented
3#79303C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#70774C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10238—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940—21no—
11C2×#932—21no—
12C2×#930—22no—
13#5477C2211yescharacteristic
14C2×#874C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5019—21no—
18#5016—22no—
19#5015—21no—
20#4988—21no—
21#4966—21no—
22#4960—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4924C2211yescharacteristic
25#4843—22no—
26#4732—22no—
27#4695—21no—
28#4694—21no—
29C3×D4⋊C8—22no—
30#4485—22no—
31#4470—22no—
32#3205—31no—
33#3036—31no—
34#2541—31no—
35#1545—31no—
36#1502—31no—
37#1478D311yescharacteristic, complemented
38#1303—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—24no—
42C2×#235—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12D412yes—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53#940—42no—
54#940—42no—
55C2×#214—22no—
56#932—42no—
57#932—42no—
58C2×#213—22no—
59#930—21no—
60#930—21no—
61#930—48no—
62C2×(C2×D3)⋊C4—24no—
63C2×C4⋊Dic3—21no—
64C2×C4×Dic3—21no—
65C2×#208—21no—
66C2×D24—24no—
67#875—21no—
68#874—21no—
69#874—21no—
70#874—21no—
71#874—21no—
72#874D411yescharacteristic, complemented
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic, complemented
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yescomplemented
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C4×C8—21no—
81C3×C4×C8—21no—
82C3⋊#95D412yes—
83#821—24no—
84#810—22no—
85#809—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yescomplemented
88C2×AGL(1,ℤ/8)—61no—
89#532—31no—
90C2×#127—31no—
91C2×#127—61no—
92C2×#121—62no—
93C2×C4×D4C2×D311yescharacteristic
94#493—61no—
95#486—61no—
96#463—62no—
97#461—61no—
98#459—61no—
99#459—62no—
100#447—62no—
101#431C2×D311yescharacteristic
102#431C2×D311yescharacteristic
103#423—31no—
104#417—61no—
105#392—62no—
106#390—61no—
107C4×C4⋊C4—31no—
108#331—62no—
109#327—62no—
110D4⋊C8—22no—
111C24×C3—21no—
112C23×D3—22no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4D812yescomplemented
119C22×C3×C4—21no—
120C22×C3×C4—44no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—48no—
127C2×#67—48no—
128C2×#67—41no—
129C2×#67—84no—
130C22×Dic3—22no—
131#235—41no—
132#235—41no—
133#235—41no—
134#235—41no—
135C2×D12—21no—
136C2×D12—22no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—22no—
140C2×D12—44no—
141C2×D12—41no—
142C2×D12—41no—
143C2×D12—41no—
144C2×D12—41no—
145C2×D12—41no—
146C2×D12—41no—
147C2×D12—41no—
148C2×D12—41no—
149C2×D12—82no—
150C2×C4×D3—41no—
151C2×Dic6—21no—
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#214—44no—
163#213—44no—
164(C2×D3)⋊C4—48no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—24no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172#208—41no—
173#208—41no—
174C2×C3⋊C8—22no—
175D24—84no—
176C2×#41—61no—
177C22×D4—32no—
178C22×D4—64no—
179C22×D4—61no—
180C22×D4—61no—
181C22×D4D1211yescharacteristic
182C23×C4—21no—
183AGL(1,ℤ/8)—122no—
184AGL(1,ℤ/8)—122no—
185C2×SD16—62no—
186C2×D8—64no—
187C2×D8—62no—
188C2×M16—61no—
189C4⋊Q8—31no—
190#127—31no—
191#127—61no—
192#127—61no—
193#127—61no—
194#127—61no—
195#127—122no—
196#127—122no—
197#124—61no—
198#121—61no—
199#121—61no—
200#121—128no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—61no—
206C4×D4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4D1211yescharacteristic
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—62no—
218#102—62no—
219#102—64no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224#95—32no—
225C23×C3—22no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3#12011yescharacteristic
231C22×D3—44no—
232C22×D3—44no—
233C22×D3—41no—
234C22×D3—42no—
235C22×D3—42no—
236C22×D3—42no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—41no—
240C3×D4—41no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—82no—
246C2×C3×C4—82no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×C3×C4C2×D811yescharacteristic
254C2×C3×C4#13511yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257#67—816no—
258#67—81no—
259#67—81no—
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264D12—41no—
265D12—41no—
266D12—42no—
267D12—44no—
268D12—81no—
269D12—82no—
270D12—82no—
271D12—82no—
272D12—81no—
273D12—81no—
274D12—81no—
275D12—81no—
276C4×D3—81no—
277C4×D3—81no—
278Dic6—41no—
279Dic6—41no—
280C3×C8—42no—
281C3×C8—81no—
282C3⋊C8—42no—
283C24—21no—
284C24—62no—
285#41—121no—
286#41—121no—
287#41—121no—
288#41—121no—
289C2×Q8—61no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—124no—
301C2×D4—128no—
302C2×D4—128no—
303C2×D4—121no—
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×D4D2412yescomplemented
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—62no—
320C22×C4—121no—
321C22×C4C2×D1211yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324SD16—124no—
325D8—124no—
326D8—244no—
327M16—121no—
328M16—121no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—62no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—64no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—128no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C42—121no—
343C42—121no—
344C42#23511yescharacteristic
345C42D3×D411yescharacteristic
346C42D3×D411yescharacteristic
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#39211yescharacteristic
356C22×C3#44711yescharacteristic
357C22×C3#44911yescharacteristic
358C2×D3—84no—
359C2×D3—84no—
360C2×D3—81no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—81no—
364C2×D3—82no—
365C2×D3—81no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—81no—
370C3×C4#44711yescharacteristic
371C3×C4#44911yescharacteristic
372C3×C4#45312yes—
373C3×C4#46611yescharacteristic
374C3×C4#46911yescharacteristic
375Dic3—44no—
376Dic3—81no—
377C23—22no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—124no—
383C23—124no—
384C23—121no—
385C23—122no—
386C23—122no—
387C23—122no—
388C23#88211yescharacteristic
389Q8—121no—
390Q8—121no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395D4—122no—
396D4—124no—
397D4—121no—
398D4—121no—
399D4—242no—
400D4—241no—
401D4—242no—
402D4—242no—
403D4—2416no—
404D4—241no—
405D4—241no—
406D4—241no—
407D4—241no—
408D4—241no—
409D4—241no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—64no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—121no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4#88211yescharacteristic
423C2×C4#88411yescharacteristic
424C2×C4#88411yescharacteristic
425C2×C4C2×D2411yescharacteristic
426C2×C4#90411yescharacteristic
427C2×C4#90811yescharacteristic
428C2×C4#90811yescharacteristic
429C2×C4#93011yescharacteristic
430C2×C4#93711yescharacteristic
431C2×C4#94011yescharacteristic
432C8—42no—
433C8—81no—
434C8—122no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—82no—
438C2×C3#199911yescharacteristic
439C2×C3#200211yescharacteristic
440C2×C3#200811yescharacteristic
441D3—161no—
442D3—162no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—42no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—244no—
451C22—244no—
452C22—242no—
453C22—241no—
454C22—241no—
455C22—242no—
456C22—242no—
457C22—241no—
458C22#473211yescharacteristic
459C22#473411yescharacteristic
460C22#496011yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—81no—
465C4—124no—
466C4—241no—
467C4#473211yescharacteristic
468C4#473411yescharacteristic
469C4#511511yescharacteristic
470C4#513511yescharacteristic
471C4#519812yes—
472C3#1023811yescharacteristic, complemented, Sylow
473C2—21no—
474C2—21no—
475C2—82no—
476C2—482no—
477C2—481no—
478C2#6996111yescharacteristic
479C2#7032411yescharacteristic
480C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
hypercenter#1478D3
hypocenterC3#10238
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #82378C2semidirect
2. #82307C2semidirect
3. #79303C2semidirect
4. C3×#1478C2semidirect
5. #70789C2semidirect
6. #70774C2semidirect
7. #67746C2semidirect
37. #1478D3semidirect
72. #874D4semidirect
75. #869D4semidirect
77. C3×C4⋊C8D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
118. C2×C3×D4D8semidirect
216. C4⋊C8#67semidirect
314. C2×D4D24semidirect
472. C3#10238semidirect

Maximal subgroups

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

subgroupindex
#823782
#823072
#793032
C3×#14782
#707892
#707742
#677462
#102383