Finite Group Databasebeta

← #10691766

Subgroups — #10691766

Order 768.

564 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 109 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#73473C211yesmaximal, characteristic, complemented
3#69946C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic
5#69456C211yesmaximal, characteristic
6C3×#1282C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9278—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5212C2211yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16#5114—22no—
17C2×#806—21no—
18C2×#806—21no—
19#4946—21no—
20Dic3×C22⋊C4—21no—
21#4933—21no—
22C4×Dic3⋊C4—21no—
23#4738—21no—
24#4735—21no—
25#4731C412yescomplemented
26#4721C412yescomplemented
27#4654—21no—
28#4646—21no—
29#4470—21no—
30#4470—21no—
31#4467—22no—
32#3187—31no—
33#2368—31no—
34#1964—31no—
35#1520—31no—
36C2×#327—31no—
37#1294—31no—
38#1282D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C22×D12C2×C411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×#217—21no—
50C2×#217—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59#924—21no—
60#924—21no—
61#924C2×C412yes—
62#924D412yes—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#204—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8C2×C412yes—
73C3×#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
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⋊C8C2×D311yescharacteristic
97C2×C22⋊C8C2×D311yescharacteristic
98#391—61no—
99#387—61no—
100#387—61no—
101#385—61no—
102#380—31no—
103C4×C4⋊C4—61no—
104C4×C22⋊C4—61no—
105C4×C22⋊C4C2×D311yescharacteristic
106C2×#95—31no—
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—81no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—42no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C22×Dic3—81no—
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—41no—
145C2×C3×C8—81no—
146C2×C3×C8—81no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4SD1612yescomplemented
151C3×C42—81no—
152#217—22no—
153#217—22no—
154#217—41no—
155#217—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—81no—
162C4⋊Dic3C22⋊C412yes—
163Dic3⋊C4—22no—
164Dic3⋊C4—22no—
165Dic3⋊C4—42no—
166Dic3⋊C4—81no—
167C4×Dic3—42no—
168C4×Dic3—81no—
169#204—82no—
170C22×Q8—31no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C22×C8—21no—
177C22×C8—21no—
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—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4D1211yescharacteristic
196C2×C22⋊C4D1211yescharacteristic
197C2×C42—21no—
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—21no—
207C22⋊C8—21no—
208C22⋊C8C4×D312yes—
209#95—21no—
210#95—31no—
211#95—31no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
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×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248#67—81no—
249#67—81no—
250#67—81no—
251#67—81no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—82no—
268C2×Dic3—82no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272D12—42no—
273D12—81no—
274Dic6—42no—
275Dic6—81no—
276C3×C8—42no—
277C3×C8—81no—
278C24—61no—
279C24C2×D1211yescharacteristic
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—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—81no—
300C22×C4—122no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—241no—
305C22×C4C2×C4×D311yescharacteristic
306C22×C4#23511yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4#23911yescharacteristic
311SD16—242no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—81no—
315C2×C8—81no—
316C4⋊C4—32no—
317C4⋊C4—62no—
318C4⋊C4—62no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—122no—
323C4⋊C4—241no—
324C4⋊C4—241no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4#20412yescomplemented
335C42—81no—
336C42—122no—
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—
366Dic3—42no—
367Dic3—42no—
368Dic3—42no—
369Dic3—81no—
370Dic3—81no—
371Dic3—161no—
372C23—41no—
373C23—41no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—241no—
378C23—241no—
379C23C4×D1211yescharacteristic
380C23#88211yescharacteristic
381C23#89511yescharacteristic
382C23C2×#20411yescharacteristic
383C23#90411yescharacteristic
384C23#90811yescharacteristic
385C23#90911yescharacteristic
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—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—242no—
422C2×C4—242no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4—241no—
426C2×C4D3×C22⋊C411yescharacteristic
427C2×C4#88411yescharacteristic
428C2×C4#89311yescharacteristic
429C2×C4#89411yescharacteristic
430C8—42no—
431C8—81no—
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
458C22C4×#20411yescharacteristic
459C22#471011yescharacteristic
460C22#473411yescharacteristic
461C22#473811yescharacteristic
462C22#484511yescharacteristic
463C22#484811yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—81no—
470C4—122no—
471C4—122no—
472C4—122no—
473C4—241no—
474C4—241no—
475C4—481no—
476C3#927811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—481no—
484C2#6868611yescharacteristic
485C2#6896411yescharacteristic
486C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9278
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. #73473C2semidirect
3. #69946C2semidirect
6. C3×#1282C2semidirect
7. #67675C2semidirect
25. #4731C4semidirect
26. #4721C4semidirect
38. #1282D3semidirect
150. C3×C22⋊C4SD16semidirect
334. C22⋊C4#204semidirect
476. C3#9278semidirect

Maximal subgroups

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

subgroupindex
#786122
#734732
#699462
C2×#44702
#694562
C3×#12822
#676752
#92783