Finite Group Databasebeta

← #10697666

Subgroups — #10697666

Order 768.

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

normal subgroups only

This group has 580 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×#3077C211yesmaximal, characteristic, complemented
2#83363C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6#70498C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10256—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#478—22no—
14C3×#476—21no—
15C3×#474—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#418—21no—
19C2×#830—22no—
20#5084—21no—
21#5083—21no—
22#5048—22no—
23#5027—21no—
24#5020—21no—
25#5019—21no—
26#5016—22no—
27#5015—21no—
28#4966—21no—
29#4924C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32#4535—22no—
33#4483—22no—
34#3246—31no—
35#3077D311yescharacteristic, complemented
36#3036—31no—
37#1954—31no—
38#1493—31no—
39#1492—31no—
40#1292—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—22no—
61C2×#214—22no—
62#932—42no—
63#932—42no—
64C2×#213—22no—
65#930—21no—
66#930—21no—
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8D412yes—
70#875—21no—
71#874D411yescharacteristic
72C4×D12—21no—
73C4×D12—21no—
74#869D411yescharacteristic
75C3×#107—22no—
76C3×#106—22no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8D412yes—
79C3×#102—22no—
80C3×#102—22no—
81#835—22no—
82#830—44no—
83#810—22no—
84#809—22no—
85#809—22no—
86#808—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yes—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×AGL(1,ℤ/8)—61no—
94#533C2×D311yescharacteristic
95#532—31no—
96C2×C4×D4C2×D311yescharacteristic
97#478—22no—
98#476—21no—
99#476—61no—
100#475—61no—
101#474—21no—
102#465—62no—
103#463—62no—
104#461—61no—
105#461—61no—
106#460—61no—
107#459—61no—
108#428—21no—
109#423—31no—
110#423C2×D311yescharacteristic
111C2×C4⋊C8—31no—
112#418—21no—
113#417—61no—
114C2×C22⋊C8—62no—
115C2×C4×C8—31no—
116Q8⋊C8—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4SD1612yescomplemented
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—41no—
136#235—41no—
137#235—41no—
138#235—41no—
139#235—41no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×C4×D3—41no—
144C2×Dic6—21no—
145C3×M16—41no—
146C3×M16—41no—
147C2×C3×C8—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164#215—82no—
165#214—44no—
166#213—44no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—42no—
179C2×#41—61no—
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—21no—
188C2×M16—61no—
189C22×C8—32no—
190C22×C8—61no—
191C4⋊Q8—31no—
192C4⋊Q8#6711yescharacteristic
193C4⋊Q8#6711yescharacteristic
194#127—31no—
195#125—21no—
196#124—61no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199#121—61no—
200#121—61no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—61no—
208C4×D4—61no—
209#117—21no—
210C2×C4⋊C4#6711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D311yescharacteristic
213#107—22no—
214#106—22no—
215#106—62no—
216C4⋊C8—21no—
217C4⋊C8—32no—
218C4⋊C8—61no—
219C4⋊C8—61no—
220C4⋊C8—61no—
221C4⋊C8#6712yes—
222#103—62no—
223#103—62no—
224#102—22no—
225#102—22no—
226#102—62no—
227#102—62no—
228C22⋊C8—124no—
229C4×C8—61no—
230C4×C8—61no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3#12111yescharacteristic
237C22×D3—41no—
238C3×Q8—42no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—41no—
244C3×D4—41no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—82no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12711yescharacteristic
261C2×C3×C4C2×SD1611yescharacteristic
262C2×C3×C4#13511yescharacteristic
263C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
264C2×C3×C4#13711yescharacteristic
265#67—81no—
266#67—81no—
267C2×Dic3—41no—
268D12—42no—
269D12—41no—
270D12—41no—
271D12—41no—
272D12—41no—
273C4×D3—81no—
274C4×D3—81no—
275Dic6—41no—
276Dic6—41no—
277C3×C8—42no—
278C3⋊C8—42no—
279C3⋊C8—42no—
280C3⋊C8—42no—
281C24—21no—
282#41—121no—
283#41—121no—
284#41—121no—
285#41—121no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—61no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—121no—
298C2×D4#21512yescomplemented
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—121no—
305C22×C4D3×D411yescharacteristic
306C22×C4#23711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308SD16—124no—
309SD16—242no—
310D8—124no—
311M16—41no—
312M16—41no—
313M16—121no—
314M16—121no—
315C2×C8—22no—
316C2×C8—62no—
317C2×C8—62no—
318C2×C8—62no—
319C2×C8—121no—
320C2×C8—121no—
321C2×C8—122no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—41no—
338C42D3×D411yescharacteristic
339C42D3×D411yescharacteristic
340C42C2×#6711yescharacteristic
341C42C2×#6711yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#46511yescharacteristic
351C22×C3#46811yescharacteristic
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4#44811yescharacteristic
362C3×C4#44911yescharacteristic
363C3×C4#45011yescharacteristic
364C3×C4#45211yescharacteristic
365C3×C4#45311yescharacteristic
366C3×C4#45411yescharacteristic
367Dic3—81no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—121no—
374C23#93911yescharacteristic
375Q8—42no—
376Q8—41no—
377Q8—41no—
378Q8—121no—
379Q8—121no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4—122no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—241no—
390D4—241no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4#93711yescharacteristic
405C2×C4#93711yescharacteristic
406C2×C4#93811yescharacteristic
407C2×C4#93911yescharacteristic
408C2×C4#94011yescharacteristic
409C2×C4C2×#21511yescharacteristic
410C2×C4#94211yescharacteristic
411C2×C4#94911yescharacteristic
412C2×C4#95011yescharacteristic
413C2×C4#95311yescharacteristic
414C8—42no—
415C8—122no—
416C8—122no—
417C8—122no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—82no—
421C2×C3#200011yescharacteristic
422C2×C3#200611yescharacteristic
423C2×C3#201711yescharacteristic
424D3—161no—
425C22—21no—
426C22—21no—
427C22—44no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22—241no—
433C22—241no—
434C22—241no—
435C22#504811yescharacteristic
436C22#504911yescharacteristic
437C22#522311yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—241no—
445C4#504611yescharacteristic
446C4#504711yescharacteristic
447C4#519811yescharacteristic
448C4#520111yescharacteristic
449C4#522711yescharacteristic
450C4#523811yescharacteristic
451C3#1025611yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—82no—
455C2—481no—
456C2#7169911yescharacteristic
457C2#7209811yescharacteristic
458C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#392
prime radical#82307C2
hypercenter#3077D3
hypocenterC3#10256
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. C3×#3077C2semidirect
2. #83363C2semidirect
4. #71826C2semidirect
5. #71614C2semidirect
6. #70498C2semidirect
7. #67711C2semidirect
35. #3077D3semidirect
126. C2×C3×D4SD16semidirect
298. C2×D4#215semidirect
451. C3#10256semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833632
#823072
#718262
#716142
#704982
#677112
#102563