Finite Group Databasebeta

← #10692905

Subgroups — #10692905

Order 768.

904 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 97 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#74119C211yesmaximal, characteristic
2#73588C211yesmaximal, characteristic
3#69988C211yesmaximal, characteristic
4#69981C211yesmaximal, characteristic, complemented
5#69533C211yesmaximal, characteristic, complemented
6C3×#1284C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9459—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×D3×C22⋊C4—21no—
12C2×#875C412yescomplemented
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5211C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#823—22no—
19#4955—21no—
20#4955—21no—
21#4955—21no—
22#4939C412yescomplemented
23#4672—22no—
24#4665—22no—
25#4665—22no—
26#4467—22no—
27#4452—22no—
28#2435—31no—
29#2308—31no—
30#1783—31no—
31#1511—31no—
32#1511—31no—
33#1284—31no—
34#1284D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C24×D3D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—44no—
43C22×D12C2×C412yes—
44C22×D12D412yes—
45C22×C4×D3—22no—
46C22×Dic6C2×C411yescharacteristic, complemented
47C2×C3×M16—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2×C412yescomplemented
51C2×#217—21no—
52C2×#217—21no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×C4⋊Dic3C2×C411yescharacteristic, complemented
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×#208—21no—
67D3×C22⋊C4—44no—
68#875—44no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95D412yes—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78#823—44no—
79C23×D4—31no—
80C2×#124—32no—
81C2×C22⋊Q8—31no—
82C22×C22⋊C4—61no—
83C2×#100—62no—
84#407—31no—
85#407C2×D311yescharacteristic
86#394—62no—
87#390—32no—
88#386—62no—
89#386—62no—
90#381—31no—
91#381C2×D311yescharacteristic
92#379—61no—
93#379—61no—
94#379—61no—
95C4×C22⋊C4C2×D311yescharacteristic
96#328—62no—
97#328—62no—
98C24×C3C2×D411yescharacteristic
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—24no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—42no—
105C23×D3—42no—
106C23×D3—42no—
107C23×D3—81no—
108C23×D3—81no—
109C23×D3C22⋊C412yes—
110C23×D3C22⋊C412yes—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—42no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C2×D12—24no—
124C2×D12—24no—
125C2×D12—22no—
126C2×D12—22no—
127C2×D12—22no—
128C2×D12—22no—
129C2×D12—42no—
130C2×D12—42no—
131C2×D12—42no—
132C2×D12—42no—
133C2×D12—42no—
134C2×D12—42no—
135C2×D12—44no—
136C2×D12—44no—
137C2×C4×D3—44no—
138C2×C4×D3—82no—
139C2×C4×D3—82no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6—42no—
143C3×M16—42no—
144C2×C3×C8—41no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C42—42no—
148#217—42no—
149#217—41no—
150#217—41no—
151(C2×D3)⋊C4—44no—
152(C2×D3)⋊C4—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—82no—
155(C2×D3)⋊C4—82no—
156(C2×D3)⋊C4—82no—
157(C2×D3)⋊C4—82no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162#208—42no—
163C2×C3⋊C8—41no—
164C25—31no—
165C25—31no—
166C22×Q8—31no—
167C22×D4—32no—
168C22×D4—32no—
169C22×D4—61no—
170C22×D4—61no—
171C22×D4—61no—
172C22×D4—61no—
173C22×D4—124no—
174C23×C4—62no—
175C23×C4C22×D311yescharacteristic
176C2×M16—21no—
177C2×M16—61no—
178#124—124no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—124no—
197C2×C42C4×D312yescomplemented
198#100—124no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#95—21no—
204#95—21no—
205#95—32no—
206#95—61no—
207#95D1212yes—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3C2×C22⋊C411yescharacteristic
212C23×C3#12011yescharacteristic
213C23×C3#12311yescharacteristic
214C22×D3—22no—
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—24no—
218C22×D3—22no—
219C22×D3—22no—
220C22×D3—22no—
221C22×D3—22no—
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—88no—
234C22×D3—81no—
235C22×D3—81no—
236C22×D3—81no—
237C22×D3—81no—
238C22×D3—88no—
239C22×D3#9914yes—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263D12—44no—
264D12—44no—
265D12—44no—
266D12—44no—
267C4×D3—88no—
268Dic6—81no—
269Dic6—81no—
270C3×C8—81no—
271C3⋊C8—81no—
272C24—32no—
273C24—32no—
274C24—64no—
275C24—61no—
276C24—61no—
277C24—61no—
278C24—61no—
279C24—122no—
280C24—122no—
281C24—122no—
282C24—241no—
283C24—241no—
284C24D3×D411yescharacteristic
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—122no—
288C2×D4—64no—
289C2×D4—64no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C2×D4—124no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—124no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—124no—
309C22×C4—242no—
310C22×C4—242no—
311C22×C4C2×C4×D311yescharacteristic
312C22×C4C2×D1211yescharacteristic
313C22×C4#23511yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4#23911yescharacteristic
317M16—42no—
318M16—122no—
319C2×C8—41no—
320C2×C8—121no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—121no—
324C4⋊C4—121no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—124no—
328C22⋊C4—124no—
329C22⋊C4—124no—
330C22⋊C4—122no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—242no—
334C22⋊C4—242no—
335C22⋊C4—242no—
336C22⋊C4—242no—
337C42—42no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—44no—
346C22×C3#38611yescharacteristic
347C22×C3C2×#9912yes—
348C22×C3#45512yes—
349C2×D3—24no—
350C2×D3—44no—
351C2×D3—44no—
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—44no—
355C2×D3—44no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—82no—
360C2×D3—84no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—84no—
364C2×D3—82no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—82no—
370C3×C4—82no—
371Dic3—82no—
372Dic3—81no—
373Dic3—81no—
374C23—22no—
375C23—22no—
376C23—34no—
377C23—41no—
378C23—62no—
379C23—62no—
380C23—64no—
381C23—62no—
382C23—62no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—122no—
387C23—122no—
388C23—122no—
389C23—122no—
390C23—122no—
391C23—122no—
392C23—121no—
393C23—121no—
394C23—122no—
395C23—122no—
396C23—122no—
397C23—241no—
398C23—241no—
399C23—241no—
400C23—241no—
401C23—248no—
402C23—248no—
403C23D3×C22⋊C411yescharacteristic
404C23#88211yescharacteristic
405C23#89411yescharacteristic
406Q8—241no—
407Q8—241no—
408D4—124no—
409D4—124no—
410D4—124no—
411D4—124no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—248no—
432C2×C4C4×D1211yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4#89311yescharacteristic
435C2×C4#89511yescharacteristic
436C8—81no—
437C8—241no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3—41no—
441C2×C3#188511yescharacteristic
442C2×C3#189811yescharacteristic
443C2×C3#191111yescharacteristic
444D3—44no—
445D3—82no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—44no—
454C22—64no—
455C22—124no—
456C22—124no—
457C22—124no—
458C22—124no—
459C22—122no—
460C22—122no—
461C22—122no—
462C22—122no—
463C22—122no—
464C22—242no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—244no—
469C22—244no—
470C22#466511yescharacteristic
471C22D3×#9912yes—
472C22#482512yes—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—82no—
478C4—82no—
479C4—242no—
480C4—241no—
481C4—241no—
482C3#945911yescharacteristic, complemented, Sylow
483C2—22no—
484C2—22no—
485C2—41no—
486C2—124no—
487C2—242no—
488C2#6870511yescharacteristic
489C2#6875911yescharacteristic
490C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9459
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
4. #69981C2semidirect
5. #69533C2semidirect
6. C3×#1284C2semidirect
7. #67616C2semidirect
12. C2×#875C4semidirect
22. #4939C4semidirect
34. #1284D3semidirect
46. C22×Dic6C2×C4semidirect
50. C2×C3×C42C2×C4semidirect
60. C2×C4⋊Dic3C2×C4semidirect
197. C2×C42C4×D3semidirect
482. C3#9459semidirect

Maximal subgroups

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

subgroupindex
#741192
#735882
#699882
#699812
#695332
C3×#12842
#676162
#94593