Finite Group Databasebeta

← #10705365

Subgroups — #10705365

Order 768.

670 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 83 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#76761C211yesmaximal, characteristic
2D3×#342C211yesmaximal, characteristic, complemented
3D3×#323C211yesmaximal, characteristic, complemented
4C3×#1343C211yesmaximal, characteristic, complemented
5#67645C211yesmaximal, characteristic, complemented
6#67491C211yesmaximal, characteristic, complemented
7#67450C211yesmaximal, characteristic, complemented
8#11692—31nomaximal, Sylow
9C2×#938C411yescharacteristic, complemented
10C2×#937—21no—
11C2×D3×C22⋊C4—21no—
12C2×D3×C22⋊C4C2211yescharacteristic
13C3×#392C2211yescharacteristic
14#5223C411yescharacteristic, complemented
15#5222—21no—
16#4959—21no—
17#4956—21no—
18#4953C2211yescharacteristic
19D3×C22⋊C8—21no—
20#4674C411yescharacteristic, complemented
21#4667C411yescharacteristic, complemented
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4468—21no—
28#2617—31no—
29C2×#342—31no—
30C2×#323—31no—
31#1344—31no—
32#1344—31no—
33#1343—31no—
34#1343D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic, complemented
38C22×#67—21no—
39C22×#67—21no—
40C22×C4×D3—41no—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2311yescharacteristic
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217D411yescharacteristic
51#938—41no—
52#938—41no—
53#937—42no—
54#937—42no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—41no—
58C2×(C2×D3)⋊C4C2×C411yescharacteristic
59C2×(C2×D3)⋊C4D411yescharacteristic
60C2×C4⋊Dic3C2×C411yescharacteristic, complemented
61C2×Dic3⋊C4—21no—
62C2×D3×C8—41no—
63D3×C22⋊C4—24no—
64D3×C22⋊C4—41no—
65D3×C22⋊C4—41no—
66D3×C22⋊C4—42no—
67C3×C22⋊C8—21no—
68C3×#95C2×C411yescharacteristic, complemented
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95C2×C411yescharacteristic, complemented
72#830—21no—
73(C2×D3)⋊C8—41no—
74C2×#121—31no—
75C2×#120—61no—
76C22×C22⋊C4—31no—
77C22×C22⋊C4—61no—
78C2×C22⋊C8—61no—
79#397—31no—
80#397—31no—
81#394—61no—
82#394—61no—
83#392—31no—
84#392C2×D311yescharacteristic
85#386—61no—
86#380—31no—
87#342—31no—
88#342—61no—
89#342C2×D311yescharacteristic
90#323—31no—
91#323—61no—
92#323C2×D311yescharacteristic
93C24×C3—21no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—22no—
98C23×D3—81no—
99C23×D3—81no—
100C23×D3C22⋊C411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C23×D3D812yes—
103C23×D3SD1612yes—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—81no—
108C2×C3×D4—81no—
109C2×C3×D4—81no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—42no—
117C2×#67—42no—
118C2×#67—42no—
119C2×#67—81no—
120C2×#67—81no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—41no—
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—44no—
131C2×C4×D3—81no—
132C2×C4×D3—82no—
133C2×C4×D3—82no—
134C2×C4×D3—81no—
135C2×C3×C8—41no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—81no—
140#217—22no—
141#217—41no—
142#217—42no—
143#217—41no—
144#217—41no—
145#217—41no—
146#217—81no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—44no—
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—42no—
152C4⋊Dic3—41no—
153Dic3⋊C4—81no—
154C2×C3⋊C8—41no—
155D3×C8—82no—
156C25—31no—
157C22×D4—21no—
158C22×D4—61no—
159C22×D4—61no—
160C22×D4C4×D311yescharacteristic, complemented
161C23×C4—31no—
162C23×C4—31no—
163C23×C4—121no—
164C22×C8—121no—
165#121—121no—
166#121—121no—
167#120—122no—
168#120—122no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—64no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—121no—
184C2×C22⋊C4—121no—
185C2×C22⋊C4—122no—
186C2×C22⋊C4—121no—
187C2×C22⋊C4C22×D311yescharacteristic
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—121no—
191#95—31no—
192#95—61no—
193#95—61no—
194#95C4×D311yescharacteristic, complemented
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3C2×C22⋊C411yescharacteristic
207C23×C3C2×D811yescharacteristic
208C23×C3C2×SD1611yescharacteristic
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—22no—
214C22×D3—22no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—81no—
222C22×D3—81no—
223C22×D3—82no—
224C22×D3—82no—
225C22×D3—84no—
226C22×D3#9911yescharacteristic
227C22×D3#9911yescharacteristic
228C22×D3#10212yes—
229C22×D3#10212yes—
230C22×D3#10412yes—
231C3×D4—82no—
232C3×D4—82no—
233C3×D4—82no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244#67—84no—
245#67—82no—
246#67—82no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C4×D3—42no—
261C4×D3—81no—
262C4×D3—81no—
263C4×D3—81no—
264C4×D3—88no—
265C3×C8—81no—
266C3⋊C8—81no—
267C24—21no—
268C24—31no—
269C24—31no—
270C24—32no—
271C24—32no—
272C24—61no—
273C24—62no—
274C24—61no—
275C24—241no—
276C24—241no—
277C24D3×D411yescharacteristic
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4—122no—
285C2×D4—122no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—122no—
289C2×D4—241no—
290C2×D4—241no—
291C22×C4—21no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—124no—
302C22×C4—121no—
303C22×C4—241no—
304C22×C4—241no—
305C22×C4—242no—
306C22×C4—242no—
307C22×C4C2×C4×D311yescharacteristic
308C22×C4D3×D411yescharacteristic
309C2×C8—41no—
310C2×C8—121no—
311C2×C8—242no—
312C4⋊C4—121no—
313C4⋊C4—241no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—62no—
318C22⋊C4—81no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—122no—
322C22⋊C4—122no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—122no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22⋊C4—241no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3C2×#9911yescharacteristic
347C22×C3C2×#10211yescharacteristic
348C22×C3C2×#10411yescharacteristic
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—22no—
354C2×D3—42no—
355C2×D3—42no—
356C2×D3—44no—
357C2×D3—44no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—42no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—81no—
364C2×D3#32712yes—
365C2×D3#32712yes—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—81no—
369C3×C4—82no—
370Dic3—41no—
371Dic3—41no—
372Dic3—81no—
373Dic3—82no—
374Dic3—81no—
375C23—21no—
376C23—21no—
377C23—31no—
378C23—31no—
379C23—32no—
380C23—32no—
381C23—32no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—62no—
392C23—62no—
393C23—62no—
394C23—62no—
395C23—61no—
396C23—61no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23—122no—
402C23—122no—
403C23—241no—
404C23—241no—
405C23—242no—
406C23—242no—
407C23—244no—
408C23D3×C22⋊C411yescharacteristic
409C23D3×D811yescharacteristic
410C23D3×SD1611yescharacteristic
411D4—82no—
412D4—82no—
413D4—82no—
414D4—242no—
415D4—242no—
416D4—244no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—241no—
440C2×C4—241no—
441C2×C4—241no—
442C2×C4—248no—
443C2×C4—241no—
444C2×C4—241no—
445C8—81no—
446C8—241no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—161no—
452C2×C3C2×#32711yescharacteristic
453C2×C3#212111yescharacteristic
454C2×C3#212711yescharacteristic
455D3—22no—
456D3—22no—
457D3—42no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—32no—
463C22—32no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—62no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—122no—
482C22—122no—
483C22—122no—
484C22—124no—
485C22—124no—
486C22—121no—
487C22—121no—
488C22—241no—
489C22—242no—
490C22—242no—
491C22D3×#9911yescharacteristic
492C22D3×#10211yescharacteristic
493C22D3×#10411yescharacteristic
494C4—41no—
495C4—41no—
496C4—81no—
497C4—82no—
498C4—121no—
499C4—121no—
500C4—241no—
501C4—242no—
502C4—241no—
503C3#1169211yescharacteristic, complemented, Sylow
504C2—21no—
505C2—21no—
506C2—41no—
507C2—41no—
508C2—62no—
509C2—62no—
510C2—122no—
511C2—161no—
512C2D3×#32711yescharacteristic
513C2#6913511yescharacteristic
514C2#6915011yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1343C2
socleC22×C3C2×#99
prime radical#76761C2
cosocleC2×C3×C22⋊C4C23
hypercenter#1343D3
hypocenterC3#11692
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1343

Complements

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

NH ≅ G/Ntype
2. D3×#342C2semidirect
3. D3×#323C2semidirect
4. C3×#1343C2semidirect
5. #67645C2semidirect
6. #67491C2semidirect
7. #67450C2semidirect
9. C2×#938C4semidirect
14. #5223C4semidirect
20. #4674C4semidirect
21. #4667C4semidirect
34. #1343D3semidirect
37. C22×C3×D4C2×C4semidirect
60. C2×C4⋊Dic3C2×C4semidirect
68. C3×#95C2×C4semidirect
71. C3⋊#95C2×C4semidirect
160. C22×D4C4×D3semidirect
194. #95C4×D3semidirect
503. C3#11692semidirect

Maximal subgroups

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

subgroupindex
#767612
D3×#3422
D3×#3232
C3×#13432
#676452
#674912
#674502
#116923