Finite Group Databasebeta

← #10695914

Subgroups — #10695914

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
1C3×#3100C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic, complemented
3#82415C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9996—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#949—21no—
12C2×#930—21no—
13C2×C4×D12C2211yescharacteristic
14C3×#480—21no—
15C3×#474—21no—
16C3×#459—21no—
17C3×#447—21no—
18C3×C4×D8—22no—
19C3×#431C2211yescharacteristic
20#5232—21no—
21#5223—21no—
22#5212—21no—
23#5083—21no—
24#5072—21no—
25#5057—22no—
26#4992—21no—
27#4956—22no—
28#4936C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4537—22no—
32D4⋊(C3⋊C8)—22no—
33#4459—22no—
34#3100D311yescharacteristic, complemented
35#3036—31no—
36#2535—31no—
37#1517—31no—
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×#67—22no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C2×C3×D8—22no—
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#127D411yescharacteristic, complemented
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63#949—42no—
64#949—42no—
65C2×#215—22no—
66C2×#213—22no—
67#930—42no—
68#930—42no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×#208—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12—21no—
78C3×#107—22no—
79C3×C4⋊C8—21no—
80C3×#102—22no—
81C3×#102—22no—
82C3×C22⋊C8—21no—
83C3×C4×C8D412yescomplemented
84#835—22no—
85#832—22no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88C3⋊#95D412yes—
89#830—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)D412yescomplemented
93C4×C3⋊C8—21no—
94C2×AGL(1,ℤ/8)—61no—
95#532C2×D311yescharacteristic
96C2×#121—61no—
97C2×C4×D4—31no—
98C2×C4×D4C2×D311yescharacteristic
99#480—21no—
100#474—21no—
101#463—62no—
102#461—61no—
103#459—21no—
104#459—61no—
105#447—21no—
106C4×D8—22no—
107#431—31no—
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—62no—
118D4⋊C8—62no—
119C24×C3—21no—
120C23×D3—21no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—41no—
125C2×C3×Q8—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—42no—
132C2×C3×D4—41no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—44no—
145C2×#67—44no—
146C2×#67—82no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12D812yescomplemented
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3—82no—
158C3×D8—44no—
159C2×C3×C8—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42#4111yescharacteristic
172#217—82no—
173#215—44no—
174#213—44no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—41no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182#208—41no—
183#208—41no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×#41—21no—
187C22×D4—31no—
188C22×D4—62no—
189C22×D4#6711yescharacteristic
190C23×C4—21no—
191C23×C4—61no—
192AGL(1,ℤ/8)—122no—
193AGL(1,ℤ/8)—122no—
194C2×SD16—62no—
195C2×D8—22no—
196C2×D8—62no—
197C2×M16—61no—
198C4⋊Q8#6711yescharacteristic, complemented
199#127#6711yescharacteristic, complemented
200#124—21no—
201#121—21no—
202#121—21no—
203#121—122no—
204#121—122no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—62no—
222C2×C42C22×D311yescharacteristic
223#107—22no—
224C4⋊C8—21no—
225C4⋊C8—32no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228#103—62no—
229#102—22no—
230#102—22no—
231#102—62no—
232C22⋊C8—21no—
233C22⋊C8—61no—
234C4×C8—61no—
235C4×C8#6712yescomplemented
236#95—32no—
237#95—62no—
238#95—61no—
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3#12011yescharacteristic
248C22×D3—22no—
249C22×D3—42no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—41no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—42no—
260C3×D4—41no—
261C3×D4—41no—
262C3×D4—81no—
263C3×D4—81no—
264C3×D4—81no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12711yescharacteristic
283C2×C3×C4C2×D811yescharacteristic
284C2×C3×C4#13511yescharacteristic
285C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
286C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
287#67—88no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293D12—22no—
294D12—22no—
295D12—41no—
296D12—41no—
297C4×D3—82no—
298C4×D3—82no—
299C3×C8—42no—
300C3⋊C8—42no—
301C3⋊C8—81no—
302C24—21no—
303C24—61no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308C2×Q8—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—32no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—124no—
323C2×D4—124no—
324C2×D4—242no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—62no—
334C22×C4—61no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—242no—
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4C2×#6711yescharacteristic
341SD16—124no—
342D8—44no—
343D8—124no—
344M16—121no—
345M16—121no—
346C2×C8—22no—
347C2×C8—62no—
348C2×C8—121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—124no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—242no—
365C42D3×D411yescharacteristic
366C42#23711yescharacteristic
367C42C2×#6711yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3#39211yescharacteristic
382C22×C3#44711yescharacteristic
383C22×C3#44911yescharacteristic
384C2×D3—44no—
385C2×D3—42no—
386C2×D3—82no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4#44711yescharacteristic
396C3×C4#44911yescharacteristic
397C3×C4#46611yescharacteristic
398C3×C4#46911yescharacteristic
399Dic3—82no—
400Dic3—81no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—62no—
410C23—122no—
411C23—122no—
412C23—121no—
413C23—121no—
414C23#93711yescharacteristic
415Q8—41no—
416Q8—41no—
417D4—22no—
418D4—22no—
419D4—41no—
420D4—41no—
421D4—42no—
422D4—41no—
423D4—41no—
424D4—62no—
425D4—62no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—121no—
430D4—121no—
431D4—248no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—242no—
450C2×C4—242no—
451C2×C4D3×D811yescharacteristic
452C2×C4#91111yescharacteristic
453C2×C4#91111yescharacteristic
454C2×C4#91211yescharacteristic
455C2×C4#93711yescharacteristic
456C2×C4#93811yescharacteristic
457C2×C4#93911yescharacteristic
458C2×C4#93911yescharacteristic
459C2×C4#94011yescharacteristic
460C2×C4#95311yescharacteristic
461C8—42no—
462C8—122no—
463C8—241no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—82no—
467C2×C3—161no—
468C2×C3#199911yescharacteristic
469C2×C3#200811yescharacteristic
470C2×C3#201011yescharacteristic
471D3—82no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—124no—
485C22—122no—
486C22—242no—
487C22—241no—
488C22—241no—
489C22#515611yescharacteristic
490C22#515811yescharacteristic
491C22#522311yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—242no—
499C4—241no—
500C4#515611yescharacteristic
501C4#515711yescharacteristic
502C4#515811yescharacteristic
503C4#515911yescharacteristic
504C3#999611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—82no—
508C2—161no—
509C2—242no—
510C2#7171511yescharacteristic
511C2#7204311yescharacteristic
512C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3100C2
socleC22×C3#392
hypercenter#3100D3
hypocenterC3#9996
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C3×#3100C2semidirect
2. #83297C2semidirect
3. #82415C2semidirect
4. #71814C2semidirect
5. #71598C2semidirect
6. #70757C2semidirect
7. #67738C2semidirect
34. #3100D3semidirect
48. C3×C4⋊Q8D4semidirect
49. C3×#127D4semidirect
83. C3×C4×C8D4semidirect
92. C4⋊(C3⋊C8)D4semidirect
154. C2×D12D8semidirect
198. C4⋊Q8#67semidirect
199. #127#67semidirect
235. C4×C8#67semidirect
504. C3#9996semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#832972
#824152
#718142
#715982
#707572
#677382
#99963