Finite Group Databasebeta

← #10693003

Subgroups — #10693003

Order 768.

838 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 93 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#84096C211yesmaximal, characteristic
2#73567C211yesmaximal, characteristic, complemented
3C3×#1789C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9479—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×C3×#104—22no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#837—22no—
16#5231C412yescomplemented
17#5223C412yescomplemented
18#5220—21no—
19#5213—22no—
20#5211C2211yescharacteristic
21C4×#217C2211yescharacteristic
22#5207C2211yescharacteristic
23#4955—21no—
24#4942—21no—
25#4940—22no—
26#4669—22no—
27#4468—22no—
28#2404—31no—
29#2312—31no—
30#1789—31no—
31#1789D311yescharacteristic, complemented
32#1462—31no—
33#1284—31no—
34#1284—31no—
35C24×D3D411yescharacteristic
36C2×C3×#41—21no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—21no—
40C2×C3×#41—44no—
41C22×C3×Q8C2×C411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4C2×C411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×#67—22no—
47C23×Dic3D411yescharacteristic
48C2×C3×M16—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42D412yescomplemented
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—22no—
54#937—44no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4D412yes—
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3D412yescomplemented
66C2×#208—21no—
67C3×#104—44no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71#837—44no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—22no—
76C3⋊#95C2×C412yes—
77#830—21no—
78#830—21no—
79C22×#41C2×D311yescharacteristic
80C2×#120—61no—
81C22×C22⋊C4—31no—
82C2×#104—22no—
83C2×#104—62no—
84#407—31no—
85#407C2×D311yescharacteristic
86#393—32no—
87#392—32no—
88#388—62no—
89#388—62no—
90#386—62no—
91#386—61no—
92#381—31no—
93#380—61no—
94#379—61no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4C2×D311yescharacteristic
97#323—62no—
98C24×C3—21no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—42no—
103C23×D3—42no—
104C23×D3—81no—
105C23×D3C22⋊C412yes—
106C3×#41—24no—
107C3×#41—24no—
108C3×#41—42no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—42no—
112C3×#41—42no—
113C3×#41—42no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—42no—
117C2×C3×D4—24no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C2×#67—44no—
138C2×#67—44no—
139C2×#67—82no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—81no—
145C22×Dic3C22⋊C412yes—
146C3×M16—42no—
147C2×C3×C8—41no—
148C3×C22⋊C4—42no—
149C3×C42—42no—
150#217—42no—
151#217—42no—
152#217—82no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—22no—
156(C2×D3)⋊C4—22no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—82no—
159(C2×D3)⋊C4—82no—
160Dic3⋊C4—82no—
161C4×Dic3—42no—
162#208—42no—
163C2×C3⋊C8—41no—
164C25—31no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—44no—
170C22×Q8C4×D311yescharacteristic
171C22×D4—21no—
172C22×D4—62no—
173C22×D4C4×D311yescharacteristic
174C23×C4—21no—
175C23×C4—31no—
176C23×C4C22×D311yescharacteristic
177C2×M16—21no—
178C2×M16—61no—
179#120—124no—
180C2×C4⋊C4—62no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—32no—
183C2×C22⋊C4—32no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42—32no—
195C2×C42#6712yescomplemented
196#104—44no—
197#104—124no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—21no—
203#95—32no—
204#95—62no—
205#95—62no—
206#95—61no—
207#95—61no—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—42no—
216C23×C3C2×C22⋊C411yescharacteristic
217C23×C3#12011yescharacteristic
218C23×C3#12311yescharacteristic
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—81no—
229C22×D3—81no—
230C22×D3—88no—
231C3×Q8—22no—
232C3×Q8—22no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—44no—
240C3×D4—44no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4—44no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4C4×D411yescharacteristic
256C2×C3×C4C4×D411yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259#67—88no—
260C2×Dic3—22no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—81no—
270C3×C8—81no—
271C3⋊C8—81no—
272C24—21no—
273C24—32no—
274C24—61no—
275C24—61no—
276C24—122no—
277C24—122no—
278C24—241no—
279C24D3×D411yescharacteristic
280#41—24no—
281#41—24no—
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—124no—
299C2×D4—124no—
300C2×D4—242no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—32no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—241no—
314C22×C4C2×C4×D311yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4#23711yescharacteristic
319C22×C4C2×#6711yescharacteristic
320M16—42no—
321M16—122no—
322C2×C8—41no—
323C2×C8—121no—
324C4⋊C4—242no—
325C22⋊C4—42no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—124no—
331C22⋊C4—122no—
332C22⋊C4—122no—
333C22⋊C4—242no—
334C22⋊C4—242no—
335C22⋊C4—242no—
336C42—42no—
337C42—122no—
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—44no—
347C22×C3—42no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#38611yescharacteristic
352C22×C3#42112yes—
353C22×C3B(4,𝔽2)12yes—
354C2×D3—44no—
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—84no—
359C2×D3—84no—
360C2×D3—82no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—42no—
366C3×C4—82no—
367Dic3—42no—
368Dic3—82no—
369Dic3—82no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—41no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—62no—
379C23—62no—
380C23—62no—
381C23—122no—
382C23—122no—
383C23—122no—
384C23—122no—
385C23—122no—
386C23—122no—
387C23—241no—
388C23—241no—
389C23—248no—
390C23D3×C22⋊C411yescharacteristic
391C23#88311yescharacteristic
392C23#93711yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401D4—44no—
402D4—44no—
403D4—248no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—62no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4#67⋊C411yescharacteristic
429C2×C4#88411yescharacteristic
430C2×C4C4×#6711yescharacteristic
431C2×C4#93811yescharacteristic
432C8—81no—
433C8—241no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—41no—
437C2×C3—82no—
438C2×C3#188911yescharacteristic
439C2×C3#189711yescharacteristic
440C2×C3#191111yescharacteristic
441D3—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—124no—
456C22—122no—
457C22—122no—
458C22—122no—
459C22—242no—
460C22—244no—
461C22—244no—
462C22#482112yes—
463C22#495611yescharacteristic
464C22#503212yes—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—82no—
471C4—122no—
472C4—242no—
473C4—242no—
474C3#947911yescharacteristic, complemented, Sylow
475C2—22no—
476C2—22no—
477C2—41no—
478C2—82no—
479C2—242no—
480C2#6949911yescharacteristic
481C2#7038311yescharacteristic
482C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#386
prime radical#84096C2
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#9479
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #73567C2semidirect
3. C3×#1789C2semidirect
4. #72138C2semidirect
5. #69967C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
16. #5231C4semidirect
17. #5223C4semidirect
31. #1789D3semidirect
50. C2×C3×C42D4semidirect
65. C2×C4×Dic3D4semidirect
195. C2×C42#67semidirect
474. C3#9479semidirect

Maximal subgroups

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

subgroupindex
#840962
#735672
C3×#17892
#721382
#699672
#676822
#676162
#94793