Finite Group Databasebeta

← #10695421

Subgroups — #10695421

Order 768.

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

normal subgroups only

This group has 716 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×#4736C211yesmaximal, characteristic
2#74032C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic, complemented
4C2×#4469C211yesmaximal, characteristic, complemented
5#69501C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7C3×#1294C211yesmaximal, characteristic, complemented
8#9919—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C22×C4⋊Dic3C2211yescharacteristic
12C22×Dic3⋊C4C2211yescharacteristic
13C2×#886—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×#830C2211yescharacteristic
19C2×#821—21no—
20C2×#818—21no—
21#4994—22no—
22#4958—21no—
23#4736—22no—
24#4736—22no—
25#4672—21no—
26#4661—21no—
27#4661—22no—
28#4661—22no—
29#4529—22no—
30#4469—22no—
31#4451—22no—
32C2×#480—31no—
33#2618—31no—
34C2×#398—31no—
35#1823—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×D12D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—42no—
64C2×C4⋊Dic3—42no—
65C2×C4⋊Dic3D412yes—
66C2×C4⋊Dic3D411yescharacteristic
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C22×C3⋊C8—21no—
74#886—42no—
75#886—42no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3×#95—21no—
79C3×#95—21no—
80C3×#95D412yes—
81C3⋊#95—24no—
82C3⋊#95—22no—
83#830—21no—
84#830—21no—
85#821—24no—
86#818—24no—
87C2×#123—61no—
88C2×#121—31no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4—61no—
92#480—62no—
93#480—62no—
94C2×#107—61no—
95C2×#102—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98#398—61no—
99#398—62no—
100#398—62no—
101#396—62no—
102#394—61no—
103#386—61no—
104#380C2×D311yescharacteristic
105C2×#95—61no—
106C2×#95C2×D311yescharacteristic
107#340—62no—
108#327—62no—
109#324—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—42no—
116C22×C3×C4—42no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—42no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3D812yes—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—44no—
167C4⋊Dic3—48no—
168C4⋊Dic3—41no—
169C4⋊Dic3—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C22×D4—31no—
177C22×D4—61no—
178C23×C4—21no—
179C23×C4—31no—
180C23×C4—31no—
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C22×C8—21no—
184C22×C8—61no—
185#123—122no—
186#123—122no—
187#121—61no—
188#121—61no—
189#121—61no—
190#121—61no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4D1212yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213#107—64no—
214#102—64no—
215C22⋊C8—21no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8D1212yes—
219#95—21no—
220#95—21no—
221#95—64no—
222#95—62no—
223#95D1212yes—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3#12011yescharacteristic
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3C2×D811yescharacteristic
230C23×C3#13511yescharacteristic
231C23×C3AGL(1,ℤ/8)11yescharacteristic
232C23×C3AGL(1,ℤ/8)11yescharacteristic
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—42no—
237C22×D3—42no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—44no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256#67—81no—
257#67—81no—
258#67—81no—
259#67—81no—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—42no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281D12—42no—
282D12—81no—
283C3×C8—42no—
284C3⋊C8—81no—
285C24—61no—
286C24D3×D411yescharacteristic
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—122no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—32no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—62no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—122no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4C2×D1211yescharacteristic
323C22×C4C2×D1211yescharacteristic
324C22×C4C2×D1211yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4#23911yescharacteristic
328C2×C8—22no—
329C2×C8—42no—
330C2×C8—122no—
331C2×C8—121no—
332C4⋊C4—42no—
333C4⋊C4—62no—
334C4⋊C4—61no—
335C4⋊C4—62no—
336C4⋊C4—61no—
337C4⋊C4—128no—
338C4⋊C4—124no—
339C4⋊C4—121no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C4⋊C4—121no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—124no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—242no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3#39211yescharacteristic
360C22×C3#44711yescharacteristic
361C22×C3#44711yescharacteristic
362C22×C3#44911yescharacteristic
363C22×C3#44911yescharacteristic
364C22×C3#45612yes—
365C22×C3#45812yes—
366C22×C3#46611yescharacteristic
367C22×C3#46911yescharacteristic
368C2×D3—82no—
369C2×D3—82no—
370C2×D3—81no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—44no—
378C3×C4—82no—
379Dic3—42no—
380Dic3—82no—
381Dic3—81no—
382Dic3—81no—
383C23—22no—
384C23—22no—
385C23—121no—
386C23—122no—
387C23—121no—
388C23—121no—
389C23—122no—
390C23#88211yescharacteristic
391C23#88211yescharacteristic
392C23#89511yescharacteristic
393C23D3×D811yescharacteristic
394C23#91411yescharacteristic
395C23#91411yescharacteristic
396C23#91911yescharacteristic
397D4—122no—
398D4—241no—
399D4—241no—
400D4—241no—
401D4—241no—
402D4—241no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4#87411yescharacteristic
439C2×C4#88211yescharacteristic
440C2×C4#89511yescharacteristic
441C2×C4#89511yescharacteristic
442C8—42no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3#160912yes—
449C2×C3#161012yes—
450C2×C3#199911yescharacteristic
451C2×C3#200511yescharacteristic
452C2×C3#202211yescharacteristic
453D3—161no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—241no—
463C22—241no—
464C22—241no—
465C22—242no—
466C22—242no—
467C22#467111yescharacteristic
468C22#474212yes—
469C22#477311yescharacteristic
470C22#477311yescharacteristic
471C22#481011yescharacteristic
472C22#481011yescharacteristic
473C22#482612yes—
474C22#486511yescharacteristic
475C22#488311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—82no—
482C4—122no—
483C4—242no—
484C4—241no—
485C4—241no—
486C3#991911yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—481no—
492C2#6829712yes—
493C2#6832812yes—
494C2#6876811yescharacteristic
495C2#6888611yescharacteristic
496C2#6896611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenter#1294D3
hypocenterC3#9919
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. C2×#4661C2semidirect
4. C2×#4469C2semidirect
5. #69501C2semidirect
7. C3×#1294C2semidirect
38. #1294D3semidirect
486. C3#9919semidirect

Maximal subgroups

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

subgroupindex
C2×#47362
#740322
C2×#46612
C2×#44692
#695012
C2×#44512
C3×#12942
#99193