Finite Group Databasebeta

← #10695917

Subgroups — #10695917

Order 768.

686 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2535C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic, complemented
4#71598C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9996—31nomaximal, Sylow
9C2×C3×#121—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#397—21no—
15C3×#392—21no—
16C3×#386—22no—
17C3×#382C2211yescharacteristic
18C3×#380—21no—
19#5217—21no—
20#5019—21no—
21#5016—22no—
22#5015—21no—
23#5015—21no—
24C4×#213—22no—
25#5007—21no—
26#4967—21no—
27#4966—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32D4⋊(C3⋊C8)—22no—
33#4451—22no—
34#3100—31no—
35#3036—31no—
36#2535D311yescharacteristic, complemented
37#1517—31no—
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×D4—22no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C2×#235—21no—
47C3×#121—42no—
48C3×#121—42no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C42C2311yescharacteristic
65C2×#214—22no—
66#932—42no—
67#932—42no—
68C2×#213—22no—
69C2×#213—22no—
70#930—21no—
71#930—21no—
72C2×#208—21no—
73#875—21no—
74#874D411yescharacteristic, complemented
75C4×D12—21no—
76C4×D12—21no—
77#869D411yescharacteristic, complemented
78C3×#95—22no—
79C3×#95—21no—
80C3×#95D412yes—
81#832—22no—
82#830—21no—
83#830—21no—
84#810—22no—
85#809—22no—
86#809—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yescomplemented
92C4×C3⋊C8—21no—
93C4×C3⋊C8D412yescomplemented
94C2×AGL(1,ℤ/8)—61no—
95#532—31no—
96C2×#121—21no—
97C2×C4×D4C2×D311yescharacteristic
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#431—31no—
108#431—31no—
109#423—31no—
110#417—61no—
111#397—21no—
112#392—21no—
113#386—22no—
114#382C2×D311yescharacteristic
115#380—21no—
116#327—62no—
117D4⋊C8—62no—
118D4⋊C8—62no—
119C24×C3—21no—
120C24×C3—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—82no—
132C2×C3×D4D812yescomplemented
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—82no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C2×#67—41no—
149#235—41no—
150#235—41no—
151#235—41no—
152#235—41no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×C4×D3—41no—
157C2×Dic6—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—82no—
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173C3×C42#4111yescharacteristic
174#214—44no—
175#213—44no—
176#213—44no—
177(C2×D3)⋊C4—41no—
178(C2×D3)⋊C4—41no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181#208—41no—
182#208—41no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×#41—61no—
187C22×D4—22no—
188C22×D4#6711yescharacteristic
189C22×D4#6711yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
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—42no—
202#121—42no—
203#121—61no—
204#121—61no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4#6711yescharacteristic
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C42C22×D311yescharacteristic
223#107—62no—
224C4⋊C8—32no—
225C4⋊C8—61no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228#103—62no—
229#102—62no—
230#102—62no—
231#102—62no—
232C22⋊C8—61no—
233C22⋊C8—61no—
234C4×C8—32no—
235C4×C8—61no—
236#95—22no—
237#95—21no—
238#95#6712yes—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3#12011yescharacteristic
252C22×D3—41no—
253C3×D4—22no—
254C3×D4—22no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—88no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—82no—
273C2×C3×C4—82no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12711yescharacteristic
284C2×C3×C4C2×D811yescharacteristic
285C2×C3×C4#13511yescharacteristic
286C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
287C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
288#67—81no—
289#67—81no—
290C2×Dic3—41no—
291D12—42no—
292D12—41no—
293D12—41no—
294D12—81no—
295C4×D3—81no—
296C4×D3—81no—
297Dic6—41no—
298Dic6—41no—
299C3⋊C8—42no—
300C3⋊C8—42no—
301C3⋊C8—81no—
302C24—21no—
303C24—21no—
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—21no—
315C2×D4—21no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—82no—
323C2×D4—121no—
324C2×D4#21312yescomplemented
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—82no—
337C22×C4—121no—
338C22×C4C2×#6711yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341SD16—124no—
342D8—124no—
343D8—124no—
344M16—121no—
345M16—121no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8—121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C22⋊C4—44no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—82no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C42D3×D411yescharacteristic
366C42D3×D411yescharacteristic
367C42D3×D411yescharacteristic
368C42#23711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—42no—
372C22×C3—44no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—81no—
377C22×C3—82no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3#39211yescharacteristic
384C22×C3#44711yescharacteristic
385C22×C3#44911yescharacteristic
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4#44711yescharacteristic
397C3×C4#44911yescharacteristic
398C3×C4#46611yescharacteristic
399C3×C4#46911yescharacteristic
400Dic3—81no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—121no—
414C23#95311yescharacteristic
415Q8—121no—
416Q8—121no—
417D4—22no—
418D4—22no—
419D4—22no—
420D4—22no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—88no—
426D4—122no—
427D4—121no—
428D4—121no—
429D4—241no—
430D4—241no—
431D4—241no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—121no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4C2×#21311yescharacteristic
452C2×C4#93211yescharacteristic
453C2×C4#93711yescharacteristic
454C2×C4#93711yescharacteristic
455C2×C4#93811yescharacteristic
456C2×C4#93911yescharacteristic
457C2×C4#93911yescharacteristic
458C2×C4#94011yescharacteristic
459C2×C4#94911yescharacteristic
460C2×C4#95011yescharacteristic
461C8—122no—
462C8—122no—
463C8—241no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—82no—
467C2×C3—82no—
468C2×C3#199911yescharacteristic
469C2×C3#200811yescharacteristic
470C2×C3#201011yescharacteristic
471D3—161no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22#521711yescharacteristic
490C22#522311yescharacteristic
491C22#523811yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4—241no—
500C4#503611yescharacteristic
501C4#503711yescharacteristic
502C4#503811yescharacteristic
503C4#503911yescharacteristic
504C3#999611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—481no—
510C2#7168411yescharacteristic
511C2#7171511yescharacteristic
512C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2535C2
socleC22×C3#392
hypercenter#2535D3
hypocenterC3#9996
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C3×#2535C2semidirect
2. #82308C2semidirect
3. #82307C2semidirect
4. #71598C2semidirect
5. #71597C2semidirect
6. #69545C2semidirect
7. #67631C2semidirect
36. #2535D3semidirect
74. #874D4semidirect
77. #869D4semidirect
91. C4⋊(C3⋊C8)D4semidirect
93. C4×C3⋊C8D4semidirect
132. C2×C3×D4D8semidirect
324. C2×D4#213semidirect
504. C3#9996semidirect

Maximal subgroups

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

subgroupindex
C3×#25352
#823082
#823072
#715982
#715972
#695452
#676312
#99963