Finite Group Databasebeta

← #10695947

Subgroups — #10695947

Order 768.

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

normal subgroups only

This group has 632 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
2#82296C211yesmaximal, characteristic, complemented
3#79298C211yesmaximal, characteristic, complemented
4C3×#1478C211yesmaximal, characteristic, complemented
5#70793C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10000—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938—21no—
11C2×C4×D12C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5215—21no—
16#5213—22no—
17#5009—22no—
18#5008—21no—
19#5007—21no—
20#4987—21no—
21#4964—21no—
22#4960—21no—
23#4953—21no—
24#4936C2211yescharacteristic
25#4924C2211yescharacteristic
26#4842—21no—
27#4841—21no—
28#4731—21no—
29#4721—21no—
30C4×#204—22no—
31C3×D4⋊C8—22no—
32#4467—22no—
33#4459—22no—
34#3101—31no—
35#3077—31no—
36#2535—31no—
37#1518—31no—
38#1486—31no—
39#1478D311yescharacteristic, complemented
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54#938—42no—
55#938—42no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58#926—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63C2×#208—21no—
64C2×#204—22no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69#870—21no—
70#869D411yescharacteristic, complemented
71#869D411yescharacteristic, complemented
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79C3×C4×C8D412yescomplemented
80#832—22no—
81#832—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#821—22no—
86#817—22no—
87#816—22no—
88#808—22no—
89#807—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yescomplemented
92#533—31no—
93C2×#121—61no—
94C2×C4×D4—31no—
95C2×C4×D4C2×D311yescharacteristic
96#483—61no—
97#478—62no—
98#476—61no—
99#476—61no—
100#474—61no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431C2×D311yescharacteristic
105#431C2×D311yescharacteristic
106#428—61no—
107#423—31no—
108#418—61no—
109#397—61no—
110#392—61no—
111#386—62no—
112#382—31no—
113#380—61no—
114#328—62no—
115D4⋊C8—22no—
116D4⋊C8—62no—
117C24×C3—21no—
118C23×D3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4SD1612yescomplemented
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—82no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—44no—
133C2×#67—44no—
134C2×#67—82no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—42no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×C3×C8—22no—
149C2×C3×C8—41no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#217—44no—
159#217—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—82no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173C4×Dic3—41no—
174#208—41no—
175#208—41no—
176C2×C3⋊C8—22no—
177#204—44no—
178C22×D4—31no—
179C22×D4—62no—
180C22×D4D1211yescharacteristic
181C23×C4—21no—
182C23×C4—61no—
183C2×SD16—62no—
184C2×M16—61no—
185C4⋊Q8—31no—
186C4⋊Q8—31no—
187#125—61no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190#121—122no—
191#121—122no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202#117—61no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4D1211yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—62no—
210C2×C42C22×D311yescharacteristic
211#107—62no—
212#106—62no—
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—32no—
217C4⋊C8—61no—
218#103—62no—
219#102—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C4×C8—21no—
225C4×C8#6712yescomplemented
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3#12011yescharacteristic
235C22×D3—22no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—41no—
245C3×D4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×C3×C4C2×SD1611yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4#13711yescharacteristic
262#67—88no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271D12—22no—
272D12—22no—
273D12—41no—
274D12—41no—
275C4×D3—81no—
276C4×D3—81no—
277C4×D3—81no—
278C4×D3—81no—
279Dic6—42no—
280Dic6—81no—
281C3×C8—42no—
282C3×C8—81no—
283C3⋊C8—42no—
284C24—21no—
285C24—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—122no—
296C2×D4—124no—
297C2×D4—124no—
298C2×D4—242no—
299C2×D4#20412yescomplemented
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—62no—
304C22×C4—61no—
305C22×C4—82no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4C2×D1211yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315SD16—124no—
316M16—121no—
317M16—121no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—124no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—242no—
340C42—121no—
341C42#23511yescharacteristic
342C42D3×D411yescharacteristic
343C42D3×D411yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—82no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3#39211yescharacteristic
353C22×C3#45011yescharacteristic
354C22×C3#45211yescharacteristic
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—81no—
367C3×C4#44811yescharacteristic
368C3×C4#44911yescharacteristic
369C3×C4#46511yescharacteristic
370C3×C4#46811yescharacteristic
371Dic3—82no—
372Dic3—81no—
373Dic3—81no—
374Dic3—81no—
375C23—22no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—62no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23#88211yescharacteristic
388Q8—122no—
389Q8—241no—
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398D4—248no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—241no—
414C2×C4—241no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4#88211yescharacteristic
418C2×C4#88411yescharacteristic
419C2×C4#88411yescharacteristic
420C2×C4C2×#20411yescharacteristic
421C2×C4#90411yescharacteristic
422C2×C4#90811yescharacteristic
423C2×C4#90911yescharacteristic
424C2×C4#93011yescharacteristic
425C2×C4#93711yescharacteristic
426C2×C4#94011yescharacteristic
427C8—42no—
428C8—81no—
429C8—122no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3#200011yescharacteristic
434C2×C3#200811yescharacteristic
435C2×C3#201011yescharacteristic
436D3—82no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22—42no—
441C22—82no—
442C22—81no—
443C22—81no—
444C22—122no—
445C22—122no—
446C22—122no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22—241no—
451C22#473111yescharacteristic
452C22#473311yescharacteristic
453C22#496011yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—81no—
459C4—242no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4#473411yescharacteristic
464C4#473811yescharacteristic
465C4#511411yescharacteristic
466C4#513411yescharacteristic
467C3#1000011yescharacteristic, complemented, Sylow
468C2—21no—
469C2—21no—
470C2—82no—
471C2—242no—
472C2#6996211yescharacteristic
473C2#7033111yescharacteristic
474C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#392
prime radical#82416C2
hypercenter#1478D3
hypocenterC3#10000
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
2. #82296C2semidirect
3. #79298C2semidirect
4. C3×#1478C2semidirect
5. #70793C2semidirect
6. #70762C2semidirect
7. #67738C2semidirect
39. #1478D3semidirect
70. #869D4semidirect
71. #869D4semidirect
79. C3×C4×C8D4semidirect
91. C4⋊(C3⋊C8)D4semidirect
124. C2×C3×D4SD16semidirect
225. C4×C8#67semidirect
299. C2×D4#204semidirect
467. C3#10000semidirect

Maximal subgroups

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

subgroupindex
#824162
#822962
#792982
C3×#14782
#707932
#707622
#677382
#100003