Finite Group Databasebeta

← #10695417

Subgroups — #10695417

Order 768.

716 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 113 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#82675C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic, complemented
4C2×C3×#327C211yesmaximal, characteristic
5#69924C211yesmaximal, characteristic, complemented
6C2×#4469C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9919—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C22×C4⋊Dic3—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C22×Dic3⋊C4C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5222—21no—
17#5220—21no—
18C2×#832—21no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×#830C2211yescharacteristic
22#5033—22no—
23#5033—22no—
24#4983—21no—
25C2×#807—21no—
26#4948—22no—
27#4661—22no—
28#4661—22no—
29C3×#327—22no—
30#4551—22no—
31#4469—22no—
32C2×#480—31no—
33#2618—31no—
34C2×#398—31no—
35#1823—31no—
36C2×#327D311yescharacteristic
37C2×#324—31no—
38#1294—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×#217—22no—
54C2×#217—21no—
55C2×#217—22no—
56#935—42no—
57#935—42no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—42no—
65C2×C4⋊Dic3—42no—
66C2×C4⋊Dic3D412yes—
67C2×C4⋊Dic3D412yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—21no—
75C22×C3⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79C3×#95D412yes—
80#832—24no—
81C3⋊#95—24no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yes—
86#807—24no—
87C2×#123—61no—
88C2×#121C2×D311yescharacteristic
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—62no—
99#398—61no—
100#398—62no—
101#396—62no—
102#394—61no—
103#386—61no—
104#380—31no—
105C2×#95—61no—
106C2×#95C2×D311yescharacteristic
107#340—62no—
108#327—22no—
109#324—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—21no—
145C22×Dic3—42no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3D812yes—
153C2×C3×C8—42no—
154C2×C3×C8—41no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—44no—
160#217—41no—
161#217—41no—
162#217—82no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—48no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4⋊Dic3—42no—
169Dic3⋊C4—44no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—42no—
176C22×D4—21no—
177C22×D4D1211yescharacteristic
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—21no—
188#121—21no—
189#121—21no—
190#121—21no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—32no—
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⋊C4D1211yescharacteristic
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—21no—
217C22⋊C8—32no—
218C22⋊C8—61no—
219#95—21no—
220#95—64no—
221#95—62no—
222#95—61no—
223#95#6712yes—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3#12011yescharacteristic
232C23×C3#12011yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3C2×D811yescharacteristic
235C23×C3#13511yescharacteristic
236C23×C3AGL(1,ℤ/8)11yescharacteristic
237C23×C3AGL(1,ℤ/8)11yescharacteristic
238C3×D4—42no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—81no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12711yescharacteristic
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282C2×Dic3—81no—
283C3×C8—81no—
284C3⋊C8—42no—
285C24—21no—
286C24D3×D411yescharacteristic
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
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—62no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—62no—
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×C4#23511yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—62no—
331C2×C8—122no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—122no—
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—82no—
360C22×C3—82no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3#39211yescharacteristic
365C22×C3#44711yescharacteristic
366C22×C3#44711yescharacteristic
367C22×C3#44911yescharacteristic
368C22×C3#44911yescharacteristic
369C22×C3#45612yes—
370C22×C3#45812yes—
371C22×C3#46611yescharacteristic
372C22×C3#46911yescharacteristic
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—44no—
378C3×C4—81no—
379Dic3—42no—
380Dic3—82no—
381Dic3—82no—
382Dic3—81no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23#88211yescharacteristic
391C23#88411yescharacteristic
392C23D3×D811yescharacteristic
393C23#91111yescharacteristic
394C23#91411yescharacteristic
395C23#91611yescharacteristic
396C23#93711yescharacteristic
397D4—42no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
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—41no—
415C2×C4—41no—
416C2×C4—41no—
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—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4#88211yescharacteristic
439C2×C4#88411yescharacteristic
440C2×C4#93011yescharacteristic
441C2×C4#94011yescharacteristic
442C8—81no—
443C8—122no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—161no—
449C2×C3#160912yes—
450C2×C3#161012yes—
451C2×C3#199911yescharacteristic
452C2×C3#200511yescharacteristic
453C2×C3#202211yescharacteristic
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—81no—
463C22—81no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22#474212yes—
468C22#477311yescharacteristic
469C22#477511yescharacteristic
470C22#478111yescharacteristic
471C22#478211yescharacteristic
472C22#482612yes—
473C22#496011yescharacteristic
474C22#515611yescharacteristic
475C22#517211yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—81no—
482C4—122no—
483C4—242no—
484C4—241no—
485C4—242no—
486C3#991911yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—161no—
492C2#6829612yes—
493C2#6829712yes—
494C2#7003111yescharacteristic
495C2#7020411yescharacteristic
496C2#7038911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenterC2×#327D3
hypocenterC3#9919
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
3. C2×#4661C2semidirect
5. #69924C2semidirect
6. C2×#4469C2semidirect
7. #67612C2semidirect
486. C3#9919semidirect

Maximal subgroups

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

subgroupindex
#826752
C2×#50332
C2×#46612
C2×C3×#3272
#699242
C2×#44692
#676122
#99193