Finite Group Databasebeta

← #10691768

Subgroups — #10691768

Order 768.

564 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 107 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#78612C211yesmaximal, characteristic
2#73339C211yesmaximal, characteristic, complemented
3#69528C211yesmaximal, characteristic
4#69510C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9278—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5210—21no—
15C4×#217—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5114—22no—
19C2×#806—21no—
20C2×#806—21no—
21#4738—21no—
22#4735—21no—
23#4731C412yescomplemented
24#4721C412yescomplemented
25#4653—21no—
26#4652—21no—
27#4649—21no—
28#4646—21no—
29#4452—22no—
30#4451—21no—
31#4451—21no—
32#3187—31no—
33#2368—31no—
34#1964—31no—
35#1520—31no—
36C2×#327—31no—
37#1294D311yescharacteristic, complemented
38#1282—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×#217—21no—
48C2×#217D411yescharacteristic
49C2×#217D411yescharacteristic
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59#924C2×C412yes—
60#924D412yes—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68C2×#204—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8C2×C412yes—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95D411yescharacteristic
74C3×#95D411yescharacteristic
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#821—22no—
83#817—22no—
84#816—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8—31no—
88C2×#121—31no—
89#483—32no—
90#481—61no—
91#465—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98#391—61no—
99#387—61no—
100#387—61no—
101#385—61no—
102#380C2×D311yescharacteristic
103C4×C4⋊C4—61no—
104C4×C22⋊C4—31no—
105C4×C22⋊C4—61no—
106C2×#95C2×D311yescharacteristic
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4—81no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—81no—
140C2×Dic6—21no—
141C2×Dic6—81no—
142C2×Dic6C22⋊C412yes—
143C2×C3×C8—22no—
144C2×C3×C8—81no—
145C3×C4⋊C4—81no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—22no—
149#217—22no—
150#217—21no—
151#217—21no—
152#217—21no—
153#217SD1612yescomplemented
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3C22⋊C412yes—
160Dic3⋊C4—22no—
161Dic3⋊C4—22no—
162Dic3⋊C4—42no—
163Dic3⋊C4—81no—
164C4×Dic3—42no—
165C4×Dic3—81no—
166C4×Dic3—81no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—81no—
169#204—82no—
170C22×Q8—31no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4—21no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C22×C8—21no—
177C22×C8—61no—
178C22⋊Q8—32no—
179C22⋊Q8—32no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C42—61no—
198C2×C42—61no—
199C2×C42—61no—
200#106—62no—
201#103—62no—
202#102—62no—
203#99—122no—
204#99—122no—
205C22⋊C8—21no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8C4×D312yes—
209#95—21no—
210#95—21no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95D1211yescharacteristic
217#95D1211yescharacteristic
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3C2×SD1611yescharacteristic
224C23×C3#13511yescharacteristic
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C23×C3#13711yescharacteristic
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—81no—
231C22×D3—81no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—82no—
245C2×C3×C4—82no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12311yescharacteristic
253#67—81no—
254#67—81no—
255#67—81no—
256#67—81no—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272D12—42no—
273D12—81no—
274Dic6—42no—
275Dic6—81no—
276C3×C8—42no—
277C3⋊C8—81no—
278C24—61no—
279C24D3×D411yescharacteristic
280C2×Q8—32no—
281C2×Q8—61no—
282C2×Q8—241no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—241no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4C2×C4×D311yescharacteristic
306C22×C4C2×D1211yescharacteristic
307C22×C4#23511yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4#23911yescharacteristic
311SD16—242no—
312C2×C8—22no—
313C2×C8—81no—
314C2×C8—121no—
315C2×C8—241no—
316C4⋊C4—32no—
317C4⋊C4—62no—
318C4⋊C4—61no—
319C4⋊C4—62no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—81no—
323C4⋊C4—122no—
324C4⋊C4—241no—
325C22⋊C4—32no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—122no—
336C42—241no—
337C42—241no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3#38611yescharacteristic
349C22×C3C4×SD1611yescharacteristic
350C22×C3#44011yescharacteristic
351C22×C3#44811yescharacteristic
352C22×C3#44911yescharacteristic
353C22×C3#46011yescharacteristic
354C22×C3#46411yescharacteristic
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C2×D3—161no—
359C2×D3—161no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—42no—
365C3×C4—81no—
366C3×C4—161no—
367Dic3—42no—
368Dic3—42no—
369Dic3—42no—
370Dic3—81no—
371Dic3—81no—
372C23—41no—
373C23—41no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—241no—
378C23—241no—
379C23#88211yescharacteristic
380C23#88411yescharacteristic
381C23#89311yescharacteristic
382C23#91111yescharacteristic
383C23D3×SD1611yescharacteristic
384C23#91611yescharacteristic
385C23#91811yescharacteristic
386Q8—122no—
387Q8—241no—
388D4—122no—
389D4—241no—
390D4—241no—
391D4—241no—
392D4—241no—
393D4—241no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4C4×D1211yescharacteristic
427C2×C4D3×C22⋊C411yescharacteristic
428C2×C4#89411yescharacteristic
429C2×C4#89511yescharacteristic
430C8—42no—
431C8—241no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3#188011yescharacteristic
439C2×C3#190611yescharacteristic
440C2×C3#191311yescharacteristic
441D3—161no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—481no—
456C22—481no—
457C22#466511yescharacteristic
458C22#476211yescharacteristic
459C22#478111yescharacteristic
460C22#479811yescharacteristic
461C22#480611yescharacteristic
462C22#485211yescharacteristic
463C22#488911yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—81no—
470C4—122no—
471C4—122no—
472C4—122no—
473C4—161no—
474C4—241no—
475C4—241no—
476C3#927811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—481no—
484C2#6877811yescharacteristic
485C2#6890911yescharacteristic
486C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9278
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. #73339C2semidirect
4. #69510C2semidirect
6. C3×#1294C2semidirect
7. #67678C2semidirect
23. #4731C4semidirect
24. #4721C4semidirect
37. #1294D3semidirect
153. #217SD16semidirect
476. C3#9278semidirect

Maximal subgroups

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

subgroupindex
#786122
#733392
#695282
#695102
C2×#44512
C3×#12942
#676782
#92783