Finite Group Databasebeta

← #10695919

Subgroups — #10695919

Order 768.

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

normal subgroups only

This group has 686 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#82416C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic, complemented
3#79297C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9996—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938—21no—
11C2×#932—21no—
12#5477C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5213—22no—
17#5019—21no—
18#5016—22no—
19#5015—21no—
20#4987—21no—
21#4966—21no—
22#4960—21no—
23#4953—21no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4846—21no—
27#4843—21no—
28#4736—21no—
29#4732—21no—
30C4×D24—22no—
31C3×D4⋊C8—22no—
32#4470—22no—
33#4459—22no—
34#3100—31no—
35#3036—31no—
36#2535—31no—
37#1517—31no—
38#1486—31no—
39#1478D311yescharacteristic, complemented
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C2×#235—21no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—22no—
55#938—42no—
56#938—42no—
57C2×#214—22no—
58#932—42no—
59#932—42no—
60C2×#213—22no—
61#930—21no—
62#930—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×#208—21no—
68C2×D24—22no—
69#875—21no—
70#874D411yescharacteristic, complemented
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77#869D411yescharacteristic, complemented
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C4×C8—21no—
83C3×C4×C8D412yescomplemented
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#821—22no—
88#821—22no—
89#818—22no—
90#810—22no—
91#809—22no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)D412yescomplemented
94C2×AGL(1,ℤ/8)—61no—
95#532—31no—
96C2×#121—61no—
97C2×C4×D4—31no—
98C2×C4×D4C2×D311yescharacteristic
99#480—61no—
100#474—61no—
101#463—62no—
102#461—61no—
103#459—61no—
104#459—61no—
105#447—61no—
106C4×D8—62no—
107#431C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#423—31no—
110#417—61no—
111#397—61no—
112#392—61no—
113#386—62no—
114#382—31no—
115#380—61no—
116#327—62no—
117D4⋊C8—22no—
118D4⋊C8—62no—
119C24×C3—21no—
120C23×D3—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4D812yescomplemented
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—82no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—44no—
135C2×#67—44no—
136C2×#67—41no—
137C2×#67—82no—
138C22×Dic3—22no—
139C22×Dic3—21no—
140#235—41no—
141#235—41no—
142#235—41no—
143#235—41no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—42no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×Dic6—21no—
158C2×C3×C8—22no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168#217—44no—
169#214—44no—
170#213—44no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—82no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182#208—41no—
183#208—41no—
184C2×C3⋊C8—22no—
185D24—44no—
186C2×#41—61no—
187C22×D4—31no—
188C22×D4—62no—
189C22×D4D1211yescharacteristic
190C23×C4—21no—
191C23×C4—61no—
192AGL(1,ℤ/8)—122no—
193AGL(1,ℤ/8)—122no—
194C2×SD16—62no—
195C2×D8—62no—
196C2×D8—62no—
197C2×M16—61no—
198C4⋊Q8—31no—
199#127—31no—
200#124—61no—
201#121—61no—
202#121—61no—
203#121—122no—
204#121—122no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—61no—
210C4×D4—61no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4D1211yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—62no—
222C2×C42C22×D311yescharacteristic
223#107—62no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—32no—
227C4⋊C8—61no—
228#103—62no—
229#102—62no—
230#102—62no—
231#102—62no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C4×C8—21no—
235C4×C8#6712yescomplemented
236#95—32no—
237#95—62no—
238#95—61no—
239C23×C3—22no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3#12011yescharacteristic
245C22×D3—22no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C3×D4—22no—
254C3×D4—22no—
255C3×D4—41no—
256C3×D4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—82no—
262C2×C3×C4—82no—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12711yescharacteristic
269C2×C3×C4C2×D811yescharacteristic
270C2×C3×C4#13511yescharacteristic
271C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
272C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
273#67—88no—
274#67—81no—
275#67—81no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283D12—22no—
284D12—22no—
285D12—41no—
286D12—41no—
287D12—42no—
288D12—41no—
289D12—41no—
290D12—81no—
291C4×D3—81no—
292C4×D3—81no—
293C4×D3—81no—
294C4×D3—81no—
295C4×D3—81no—
296C4×D3—81no—
297Dic6—41no—
298Dic6—41no—
299C3×C8—42no—
300C3×C8—81no—
301C3⋊C8—42no—
302C24—21no—
303C24—61no—
304#41—121no—
305#41—121no—
306#41—121no—
307#41—121no—
308C2×Q8—61no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—122no—
320C2×D4—124no—
321C2×D4—124no—
322C2×D4—121no—
323C2×D4—242no—
324C2×D4D2412yescomplemented
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—62no—
329C22×C4—61no—
330C22×C4—82no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4C2×D1211yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341SD16—124no—
342D8—124no—
343D8—124no—
344M16—121no—
345M16—121no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—62no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—124no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—242no—
365C42#23511yescharacteristic
366C42D3×D411yescharacteristic
367C42D3×D411yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3#44711yescharacteristic
378C22×C3#44911yescharacteristic
379C2×D3—42no—
380C2×D3—42no—
381C2×D3—42no—
382C2×D3—81no—
383C2×D3—81no—
384C2×D3—81no—
385C2×D3—81no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—81no—
394C3×C4#44711yescharacteristic
395C3×C4#44911yescharacteristic
396C3×C4#46611yescharacteristic
397C3×C4#46911yescharacteristic
398Dic3—82no—
399Dic3—81no—
400Dic3—81no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—62no—
407C23—121no—
408C23—121no—
409C23—121no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23#88211yescharacteristic
415Q8—121no—
416Q8—121no—
417D4—22no—
418D4—22no—
419D4—41no—
420D4—41no—
421D4—62no—
422D4—62no—
423D4—121no—
424D4—121no—
425D4—122no—
426D4—121no—
427D4—121no—
428D4—241no—
429D4—248no—
430D4—241no—
431D4—241no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4—121no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4#88211yescharacteristic
452C2×C4#88411yescharacteristic
453C2×C4#88411yescharacteristic
454C2×C4C2×D2411yescharacteristic
455C2×C4#90411yescharacteristic
456C2×C4#90811yescharacteristic
457C2×C4#90811yescharacteristic
458C2×C4#93011yescharacteristic
459C2×C4#93711yescharacteristic
460C2×C4#94011yescharacteristic
461C8—42no—
462C8—81no—
463C8—122no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—82no—
467C2×C3#199911yescharacteristic
468C2×C3#200811yescharacteristic
469C2×C3#201011yescharacteristic
470D3—82no—
471D3—161no—
472C22—21no—
473C22—21no—
474C22—44no—
475C22—42no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—122no—
480C22—122no—
481C22—122no—
482C22—241no—
483C22—241no—
484C22—241no—
485C22—241no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22#473211yescharacteristic
490C22#473411yescharacteristic
491C22#496011yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—242no—
498C4—241no—
499C4—241no—
500C4#473211yescharacteristic
501C4#473411yescharacteristic
502C4#511511yescharacteristic
503C4#513511yescharacteristic
504C3#999611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—82no—
508C2—242no—
509C2—481no—
510C2#6996111yescharacteristic
511C2#7033111yescharacteristic
512C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
hypercenter#1478D3
hypocenterC3#9996
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. #82416C2semidirect
2. #82307C2semidirect
3. #79297C2semidirect
4. C3×#1478C2semidirect
5. #70790C2semidirect
6. #70762C2semidirect
7. #67738C2semidirect
39. #1478D3semidirect
70. #874D4semidirect
77. #869D4semidirect
83. C3×C4×C8D4semidirect
93. C4⋊(C3⋊C8)D4semidirect
126. C2×C3×D4D8semidirect
235. C4×C8#67semidirect
324. C2×D4D24semidirect
504. C3#9996semidirect

Maximal subgroups

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

subgroupindex
#824162
#823072
#792972
C3×#14782
#707902
#707622
#677382
#99963