Finite Group Databasebeta

← #10691656

Subgroups — #10691656

Order 768.

750 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#4738C211yesmaximal, characteristic
2#73876C211yesmaximal, characteristic, complemented
3#69528C211yesmaximal, characteristic
4#69525C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic
6C3×#1289C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9262—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×#204C412yescomplemented
12C2×#887—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#816C412yescomplemented
19#4973—21no—
20C2×#810—22no—
21C2×#806—21no—
22C2×#806—21no—
23#4942—21no—
24Dic3×C22⋊C4—21no—
25#4738—21no—
26#4738—22no—
27#4738—21no—
28#4646—22no—
29#4451—22no—
30C2×#448—31no—
31Q8×C22⋊C4—31no—
32#1964—31no—
33#1787—31no—
34#1519—31no—
35#1289D311yescharacteristic, complemented
36#1282—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C23×Dic3—21no—
40C22×D12C2×C411yescharacteristic
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—44no—
45C22×Dic6C2×C412yes—
46C22×Dic6D411yescharacteristic
47C22×Dic6D412yes—
48C22×Dic6D411yescharacteristic
49C22×C3×C8C2×C412yes—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×#217D411yescharacteristic
55C2×#217D411yescharacteristic
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—22no—
64C2×C4×Dic3—21no—
65C22×C3⋊C8—21no—
66C2×#204—22no—
67C2×#204—22no—
68C2×#204—44no—
69#887—44no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95—21no—
73C3×#95D412yes—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#816—22no—
80#816—22no—
81#810—44no—
82#806—42no—
83#806—42no—
84C23×Q8—31no—
85C22×SD16—32no—
86C2×#121—31no—
87C2×C4×Q8—61no—
88#448—61no—
89#448—62no—
90#448—61no—
91C2×#103—32no—
92C2×#103—62no—
93C2×#99—61no—
94C2×#99—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#391—61no—
98#391—62no—
99#380—61no—
100#380C2×D311yescharacteristic
101#380C2×D311yescharacteristic
102C4×C22⋊C4—31no—
103C4×C22⋊C4—61no—
104#327—62no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C22×C3×C4—22no—
108C22×C3×C4—42no—
109C22×C3×C4—41no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—41no—
120C2×#67—41no—
121C2×#67—41no—
122C22×Dic3—21no—
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—44no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×Dic6—21no—
134C2×Dic6—24no—
135C2×Dic6—24no—
136C2×Dic6—24no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—21no—
140C2×Dic6—42no—
141C2×Dic6—42no—
142C2×Dic6—42no—
143C2×Dic6—44no—
144C2×Dic6—44no—
145C2×Dic6—44no—
146C2×Dic6C22⋊C412yes—
147C2×Dic6C22⋊C412yes—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C3×C4⋊C4—42no—
151C3×C4⋊C4—42no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156#217—21no—
157#217—21no—
158#217—21no—
159#217SD1612yescomplemented
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164Dic3⋊C4—44no—
165C4×Dic3—44no—
166C4×Dic3—42no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—41no—
169#204—44no—
170#204—44no—
171C22×Q8—32no—
172C22×Q8—31no—
173C22×Q8—32no—
174C22×Q8—31no—
175C22×Q8—61no—
176C22×Q8—61no—
177C22×Q8—61no—
178C22×Q8—124no—
179C22×D4—31no—
180C22×D4—61no—
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×SD16—62no—
185C2×SD16—124no—
186C22×C8—61no—
187C22×C8C4×D312yes—
188#121—61no—
189#121—61no—
190#121—61no—
191#121—61no—
192C4×Q8—124no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—62no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—61no—
202C2×C42—62no—
203C2×C42—61no—
204#103—62no—
205#103—62no—
206#103—124no—
207#99—122no—
208#99—122no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#95—21no—
214#95—62no—
215#95—62no—
216#95—61no—
217#95D1212yes—
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—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
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—24no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—44no—
257C2×Dic3—44no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267D12—42no—
268D12—81no—
269Dic6—24no—
270Dic6—22no—
271Dic6—22no—
272Dic6—22no—
273Dic6—44no—
274Dic6—44no—
275Dic6—42no—
276Dic6#9914yescomplemented
277C3×C8—42no—
278C3⋊C8—81no—
279C24—61no—
280C24D3×D411yescharacteristic
281C2×Q8—32no—
282C2×Q8—32no—
283C2×Q8—64no—
284C2×Q8—64no—
285C2×Q8—64no—
286C2×Q8—62no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×Q8—62no—
290C2×Q8—122no—
291C2×Q8—122no—
292C2×Q8—122no—
293C2×Q8—124no—
294C2×Q8—124no—
295C2×Q8—124no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—122no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—121no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—124no—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4C2×D1211yescharacteristic
318C22×C4#23511yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4#23911yescharacteristic
322SD16—124no—
323SD16—124no—
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—122no—
327C2×C8—121no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—124no—
333C22⋊C4—32no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—61no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—122no—
344C42—124no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3#38611yescharacteristic
356C22×C3C4×SD1611yescharacteristic
357C22×C3#44011yescharacteristic
358C22×C3#44811yescharacteristic
359C22×C3#44911yescharacteristic
360C22×C3#46011yescharacteristic
361C22×C3#46411yescharacteristic
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—161no—
366C2×D3—161no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—82no—
371C3×C4—161no—
372C3×C4C2×#9912yes—
373Dic3—24no—
374Dic3—42no—
375Dic3—42no—
376Dic3—81no—
377Dic3—81no—
378C23—41no—
379C23—41no—
380C23—121no—
381C23—121no—
382C23—121no—
383C23—241no—
384C23—241no—
385C23#88211yescharacteristic
386C23#88411yescharacteristic
387C23#89311yescharacteristic
388C23#91111yescharacteristic
389C23D3×SD1611yescharacteristic
390C23#91611yescharacteristic
391C23#91811yescharacteristic
392Q8—34no—
393Q8—64no—
394Q8—62no—
395Q8—62no—
396Q8—62no—
397Q8—124no—
398Q8—124no—
399Q8—122no—
400D4—122no—
401D4—241no—
402D4—241no—
403D4—241no—
404D4—241no—
405D4—241no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—64no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—121no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4C4×D1211yescharacteristic
434C2×C4D3×C22⋊C411yescharacteristic
435C2×C4#89411yescharacteristic
436C2×C4#89511yescharacteristic
437C8—42no—
438C8—241no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#188011yescharacteristic
446C2×C3#189811yescharacteristic
447C2×C3#190511yescharacteristic
448D3—161no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22—481no—
463C22—481no—
464C22#466511yescharacteristic
465C22#476211yescharacteristic
466C22#478111yescharacteristic
467C22#479811yescharacteristic
468C22#480611yescharacteristic
469C22#485211yescharacteristic
470C22#488911yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—64no—
475C4—82no—
476C4—122no—
477C4—122no—
478C4—161no—
479C4—241no—
480C4—241no—
481C4D3×#9912yes—
482C3#926211yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—481no—
490C2#6870711yescharacteristic
491C2#6890911yescharacteristic
492C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9262
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #73876C2semidirect
4. #69525C2semidirect
6. C3×#1289C2semidirect
7. #67677C2semidirect
11. C22×#204C4semidirect
18. C2×#816C4semidirect
35. #1289D3semidirect
159. #217SD16semidirect
276. Dic6#99semidirect
482. C3#9262semidirect

Maximal subgroups

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

subgroupindex
C2×#47382
#738762
#695282
#695252
#695092
C3×#12892
#676772
#92623