Finite Group Databasebeta

← #10695752

Subgroups — #10695752

Order 768.

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

normal subgroups only

This group has 772 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
1#82680C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic, complemented
4C2×C3×#327C211yesmaximal, characteristic, complemented
5#69422C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9968—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#939—22no—
11C2×#930C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5223—21no—
16C2×#832—21no—
17C2×C3⋊#95—22no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#821—21no—
21C2×#817—21no—
22#5034—22no—
23#5034—22no—
24#4993—21no—
25C2×#808—21no—
26#4960—21no—
27#4949—21no—
28#4735—22no—
29#4735—22no—
30C3×#327—22no—
31#4451—22no—
32C2×#481—31no—
33C2×#481—31no—
34#2656—31no—
35C2×#327—31no—
36C2×#327D311yescharacteristic, complemented
37#1445—31no—
38#1280—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—22no—
45C23×Dic3—21no—
46C23×Dic3D412yes—
47C22×D12D411yescharacteristic
48C22×C3×C8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×#217—22no—
56#939—44no—
57#939—44no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3D412yescomplemented
71C2×Dic3⋊C4—22no—
72C22×C3⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yescomplemented
75C3×#95—21no—
76C3×#95—21no—
77#832—24no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—44no—
81#830—21no—
82#830D412yescomplemented
83#821—24no—
84#817—24no—
85#808—24no—
86C2×#121—31no—
87C2×#121—62no—
88C2×#121C2×D311yescharacteristic
89C22×C4⋊C4—31no—
90#481—62no—
91#481—62no—
92#481—62no—
93#481—62no—
94C2×#106—61no—
95C2×#106—61no—
96C2×#102—61no—
97C2×#102—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#399—61no—
101#399—61no—
102#392—61no—
103#392—61no—
104C2×#95—31no—
105C2×#95—62no—
106C2×#95C2×D311yescharacteristic
107#327—22no—
108#327—62no—
109C24×C3—21no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4—41no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—44no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—42no—
136C2×#67—42no—
137C2×#67—82no—
138C2×#67—82no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—42no—
144C22×Dic3—42no—
145C22×Dic3—42no—
146C22×Dic3—42no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3SD1614yes—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—44no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—22no—
164C4⋊Dic3—21no—
165C4⋊Dic3—22no—
166C4⋊Dic3—21no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169Dic3⋊C4—44no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—42no—
172C22×D4—21no—
173C22×D4—31no—
174C22×D4—61no—
175C22×D4—62no—
176C22×D4D1211yescharacteristic
177C23×C4—21no—
178C23×C4—32no—
179C23×C4—61no—
180C23×C4C22×D311yescharacteristic
181C22×C8—21no—
182C22×C8—61no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187#121—61no—
188#121—61no—
189#121—61no—
190#121—61no—
191#121—124no—
192#121—124no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4D1211yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206#106—64no—
207#106—64no—
208#102—64no—
209#102—64no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8#6712yescomplemented
214#95—21no—
215#95—21no—
216#95—61no—
217#95—61no—
218#95—124no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C23×C3C2×SD1612yes—
231C23×C3AGL(1,ℤ/8)12yes—
232C22×D3—41no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—42no—
237C3×D4—42no—
238C3×D4—81no—
239C3×D4—82no—
240C3×D4—81no—
241C3×D4—81no—
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—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—82no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259#67—88no—
260#67—82no—
261#67—81no—
262#67—81no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—82no—
272C2×Dic3—82no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275D12—42no—
276D12—81no—
277C3×C8—42no—
278C3⋊C8—42no—
279C24—21no—
280C24—61no—
281C24D3×D411yescharacteristic
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—122no—
292C2×D4—124no—
293C2×D4—121no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—121no—
297C2×D4—242no—
298C2×D4—242no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—34no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—61no—
310C22×C4—62no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4C2×D1211yescharacteristic
318C22×C4#23511yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C2×C8—22no—
324C2×C8—42no—
325C2×C8—62no—
326C2×C8—122no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—62no—
330C4⋊C4—61no—
331C4⋊C4—62no—
332C4⋊C4—61no—
333C4⋊C4—124no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3#39211yescharacteristic
356C22×C3#45012yes—
357C22×C3#45311yescharacteristic
358C22×C3#45411yescharacteristic
359C22×C3#45812yes—
360C22×C3#46012yes—
361C22×C3#49211yescharacteristic
362C22×C3#49611yescharacteristic
363C2×D3—82no—
364C2×D3—82no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—82no—
373C3×C4—81no—
374Dic3—44no—
375Dic3—82no—
376Dic3—81no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—121no—
386C23—122no—
387C23—122no—
388C23—121no—
389C23—121no—
390C23#88211yescharacteristic
391C23#88411yescharacteristic
392C23#91112yes—
393C23D3×SD1612yes—
394C23#94011yescharacteristic
395D4—42no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—82no—
400D4—122no—
401D4—241no—
402D4—248no—
403D4—241no—
404D4—241no—
405D4—242no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—64no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—121no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—242no—
428C2×C4—242no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4#88211yescharacteristic
432C2×C4#88411yescharacteristic
433C2×C4#93011yescharacteristic
434C2×C4#93711yescharacteristic
435C8—42no—
436C8—122no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—161no—
441C2×C3#161312yes—
442C2×C3#161612yes—
443C2×C3#200611yescharacteristic
444C2×C3#201411yescharacteristic
445C2×C3#202211yescharacteristic
446D3—161no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—81no—
456C22—81no—
457C22—82no—
458C22—82no—
459C22—81no—
460C22—241no—
461C22—241no—
462C22—242no—
463C22—242no—
464C22—241no—
465C22#474212yes—
466C22#476212yes—
467C22#477612yes—
468C22#482211yescharacteristic
469C22#482711yescharacteristic
470C22#496011yescharacteristic
471C22#515411yescharacteristic
472C22#517511yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—82no—
478C4—81no—
479C4—124no—
480C4—241no—
481C4—242no—
482C3#996811yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—22no—
486C2—161no—
487C2—481no—
488C2#6829012yes—
489C2#6829912yes—
490C2#7003111yescharacteristic
491C2#7021011yescharacteristic
492C2#7039211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radical#82680C2
hypercenterC2×#327D3
hypocenterC3#9968
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. C2×#4735C2semidirect
4. C2×C3×#327C2semidirect
5. #69422C2semidirect
6. C2×#4451C2semidirect
7. #67674C2semidirect
36. C2×#327D3semidirect
70. C2×C4⋊Dic3D4semidirect
74. C3×C22⋊C8D4semidirect
82. #830D4semidirect
213. C22⋊C8#67semidirect
482. C3#9968semidirect

Maximal subgroups

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

subgroupindex
#826802
C2×#50342
C2×#47352
C2×C3×#3272
#694222
C2×#44512
#676742
#99683