Finite Group Databasebeta

← #10697656

Subgroups — #10697656

Order 768.

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

normal subgroups only

This group has 634 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#83310C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6#70498C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10254—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#463—22no—
15C3×#461—21no—
16C3×#459—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19C2×#830—22no—
20#5087—21no—
21#5072—21no—
22#5038—22no—
23#5019—21no—
24#5016—22no—
25#5015—21no—
26#5015—21no—
27#5007—21no—
28#4966—21no—
29#4924C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4483—22no—
34#3245—31no—
35#3036—31no—
36#3036D311yescharacteristic, complemented
37#1954—31no—
38#1492—31no—
39#1492—31no—
40#1292—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×#235—21no—
45C3×AGL(1,ℤ/8)—42no—
46C3×AGL(1,ℤ/8)—42no—
47C2×C3×SD16—22no—
48C2×C3×D8—22no—
49C2×C3×M16—21no—
50C3×C4⋊Q8D411yescharacteristic
51C3×#127D411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#214—22no—
65#932—42no—
66#932—42no—
67C2×#213—22no—
68C2×#213—22no—
69#930—21no—
70#930—21no—
71C2×#208—21no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8D412yes—
74#875—21no—
75#874D411yescharacteristic
76C4×D12—21no—
77C4×D12—21no—
78#869D411yescharacteristic
79C3×C4⋊C8—21no—
80C3×C4⋊C8D412yes—
81C3×#103—22no—
82C3×#102—22no—
83#832—22no—
84#830—44no—
85#810—22no—
86#809—22no—
87#809—22no—
88#807—22no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)D412yes—
93C4×C3⋊C8—21no—
94C4×C3⋊C8—21no—
95C2×AGL(1,ℤ/8)—21no—
96C2×AGL(1,ℤ/8)—61no—
97#532—31no—
98#532C2×D311yescharacteristic
99C2×C4×D4C2×D311yescharacteristic
100#474—61no—
101#474—61no—
102#466—62no—
103#463—22no—
104#463—62no—
105#461—21no—
106#461—61no—
107#459—21no—
108#459—61no—
109#459—61no—
110#459—61no—
111#423—31no—
112#423C2×D311yescharacteristic
113C2×C4⋊C8—31no—
114#417—21no—
115#417—61no—
116C2×C22⋊C8—62no—
117C2×C4×C8—31no—
118D4⋊C8—62no—
119D4⋊C8—62no—
120C24×C3—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—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—41no—
135C2×C3×D4D812yescomplemented
136C22×C3×C4—21no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—41no—
146#235—41no—
147#235—41no—
148#235—41no—
149#235—41no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×C4×D3—41no—
154C2×Dic6—21no—
155C3×SD16—44no—
156C3×D8—44no—
157C3×M16—41no—
158C3×M16—41no—
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×C42C2×D411yescharacteristic
172#214—44no—
173#213—44no—
174#213—82no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177C4⋊Dic3—21no—
178C4⋊Dic3—21no—
179#208—41no—
180#208—41no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—42no—
187C2×#41—21no—
188C2×#41—61no—
189C22×D4#6711yescharacteristic
190C23×C4—21no—
191AGL(1,ℤ/8)—42no—
192AGL(1,ℤ/8)—42no—
193AGL(1,ℤ/8)—122no—
194AGL(1,ℤ/8)—122no—
195C2×SD16—22no—
196C2×SD16—62no—
197C2×D8—22no—
198C2×D8—62no—
199C2×D8—62no—
200C2×M16—21no—
201C2×M16—61no—
202C22×C8—32no—
203C22×C8—61no—
204C4⋊Q8—31no—
205C4⋊Q8#6711yescharacteristic
206#127—31no—
207#127#6711yescharacteristic
208#124—21no—
209#124—61no—
210#121—21no—
211#121—21no—
212#121—61no—
213#121—61no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—61no—
221C4×D4—61no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C42C22×D311yescharacteristic
225#107—62no—
226C4⋊C8—21no—
227C4⋊C8—32no—
228C4⋊C8—61no—
229C4⋊C8—61no—
230C4⋊C8—61no—
231C4⋊C8#6712yes—
232#103—22no—
233#103—62no—
234#102—22no—
235#102—62no—
236#102—62no—
237#102—62no—
238C22⋊C8—124no—
239C4×C8—61no—
240C4×C8—61no—
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3#12111yescharacteristic
248C22×D3—41no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×D4—22no—
252C3×D4—22no—
253C3×D4—42no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—81no—
261C3×D4—81no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—82no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4#12011yescharacteristic
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12711yescharacteristic
279C2×C3×C4C2×D811yescharacteristic
280C2×C3×C4#13511yescharacteristic
281C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
282C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
283#67—81no—
284#67—81no—
285C2×Dic3—41no—
286D12—42no—
287D12—41no—
288D12—41no—
289D12—41no—
290D12—41no—
291C4×D3—81no—
292C4×D3—81no—
293Dic6—41no—
294Dic6—41no—
295C3×C8—42no—
296C3⋊C8—42no—
297C3⋊C8—42no—
298C3⋊C8—42no—
299C24—21no—
300#41—41no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—121no—
305#41—121no—
306#41—121no—
307#41—121no—
308C2×Q8—21no—
309C2×Q8—61no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—121no—
323C2×D4#21312yescomplemented
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—121no—
331C22×C4D3×D411yescharacteristic
332C22×C4#23711yescharacteristic
333C22×C4C2×#6711yescharacteristic
334SD16—44no—
335SD16—124no—
336D8—44no—
337D8—124no—
338D8—242no—
339M16—41no—
340M16—41no—
341M16—121no—
342M16—121no—
343C2×C8—22no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—121no—
348C2×C8—121no—
349C2×C8—122no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C42D3×D411yescharacteristic
363C42D3×D411yescharacteristic
364C42C2×#6711yescharacteristic
365C42C2×#6711yescharacteristic
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—44no—
369C22×C3—42no—
370C22×C3—82no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3#46611yescharacteristic
378C22×C3#46911yescharacteristic
379C2×D3—81no—
380C2×D3—81no—
381C2×D3—81no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—81no—
386C3×C4—81no—
387C3×C4#44711yescharacteristic
388C3×C4#44711yescharacteristic
389C3×C4#44911yescharacteristic
390C3×C4#44911yescharacteristic
391C3×C4#45311yescharacteristic
392C3×C4#45411yescharacteristic
393Dic3—81no—
394C23—22no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—121no—
401C23#93911yescharacteristic
402Q8—41no—
403Q8—41no—
404Q8—121no—
405Q8—121no—
406D4—22no—
407D4—22no—
408D4—42no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—81no—
416D4—81no—
417D4—122no—
418D4—121no—
419D4—121no—
420D4—121no—
421D4—121no—
422D4—241no—
423D4—241no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—121no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4C2×#21311yescharacteristic
439C2×C4#93211yescharacteristic
440C2×C4#93711yescharacteristic
441C2×C4#93711yescharacteristic
442C2×C4#93811yescharacteristic
443C2×C4#93911yescharacteristic
444C2×C4#94011yescharacteristic
445C2×C4#94911yescharacteristic
446C2×C4#95011yescharacteristic
447C2×C4#95311yescharacteristic
448C8—42no—
449C8—122no—
450C8—122no—
451C8—122no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—82no—
455C2×C3—161no—
456C2×C3#199911yescharacteristic
457C2×C3#200611yescharacteristic
458C2×C3#201711yescharacteristic
459D3—161no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—42no—
464C22—82no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22#503811yescharacteristic
474C22#503911yescharacteristic
475C22#522311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—241no—
482C4#503611yescharacteristic
483C4#503711yescharacteristic
484C4#519811yescharacteristic
485C4#520111yescharacteristic
486C4#521711yescharacteristic
487C4#523811yescharacteristic
488C3#1025411yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—82no—
492C2—161no—
493C2—481no—
494C2#7168411yescharacteristic
495C2#7209811yescharacteristic
496C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10254
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. #83310C2semidirect
3. #82307C2semidirect
4. #71802C2semidirect
5. #71614C2semidirect
6. #70498C2semidirect
7. #67711C2semidirect
36. #3036D3semidirect
135. C2×C3×D4D8semidirect
323. C2×D4#213semidirect
488. C3#10254semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833102
#823072
#718022
#716142
#704982
#677112
#102543