Finite Group Databasebeta

← #10695416

Subgroups — #10695416

Order 768.

716 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2618C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic
3C2×#4981C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic
5C2×#4528C211yesmaximal, characteristic, complemented
6#69501C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9919—31nomaximal, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×Dic3⋊C4—21no—
14C3×#398—21no—
15C3×#396—22no—
16C3×#394—21no—
17C3×#386—21no—
18C3×#380C2211yescharacteristic
19C2×#832—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5033—22no—
25#5033—22no—
26#4981—22no—
27#4981—22no—
28C2×#807—21no—
29#4537—22no—
30#4529—22no—
31#4528—22no—
32C2×#480—31no—
33#2618D311yescharacteristic
34C2×#398—31no—
35#1823—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C3×#123—42no—
47C3×#123—42no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D412yes—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D412yes—
71C2×Dic3⋊C4—24no—
72C2×Dic3⋊C4—42no—
73C2×Dic3⋊C4—42no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C3×#95—22no—
77C3×#95—21no—
78#832—24no—
79C3⋊#95—24no—
80C3⋊#95—21no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#807—24no—
87C2×#123—21no—
88C2×#121C2×D311yescharacteristic
89C22×C4⋊C4—31no—
90C22×C4⋊C4—61no—
91C22×C4⋊C4C2×D311yescharacteristic
92#480—62no—
93#480—62no—
94C2×#107—61no—
95C2×#102—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98#398—21no—
99#398—62no—
100#398—62no—
101#396—22no—
102#394—21no—
103#386—21no—
104#380C2×D311yescharacteristic
105C2×#95—31no—
106C2×#95—61no—
107#340—62no—
108#327—62no—
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—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4D812yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—42no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—44no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—44no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—82no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171Dic3⋊C4—48no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C22×D4—21no—
177C22×D4#6711yescharacteristic
178C23×C4—61no—
179C23×C4—61no—
180C23×C4#6711yescharacteristic
181C23×C4#6711yescharacteristic
182C23×C4C22×D311yescharacteristic
183C22×C8—61no—
184C22×C8—61no—
185#123—42no—
186#123—42no—
187#121—21no—
188#121—21no—
189#121—21no—
190#121—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—64no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—122no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C4⋊C4#6712yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213#107—64no—
214#102—64no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219#95—22no—
220#95—21no—
221#95—32no—
222#95—64no—
223#95—61no—
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—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—81no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12711yescharacteristic
271C2×Dic3—24no—
272C2×Dic3—44no—
273C2×Dic3—44no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282C2×Dic3—81no—
283C3⋊C8—42no—
284C3⋊C8—81no—
285C24—21no—
286C24C2×#6711yescharacteristic
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—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—64no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4#21312yes—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4#23711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C2×C8—62no—
329C2×C8—122no—
330C2×C8—122no—
331C2×C8—121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—62no—
341C4⋊C4—62no—
342C4⋊C4—128no—
343C4⋊C4—121no—
344C4⋊C4—121no—
345C22⋊C4—44no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—82no—
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—42no—
378C3×C4—82no—
379C3×C4—81no—
380C3×C4—81no—
381Dic3—44no—
382Dic3—82no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23C2×#21311yescharacteristic
391C23#93211yescharacteristic
392C23#93711yescharacteristic
393C23#93911yescharacteristic
394C23#94911yescharacteristic
395C23#95011yescharacteristic
396C23#95311yescharacteristic
397D4—42no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—64no—
426C2×C4—81no—
427C2×C4—122no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4#93711yescharacteristic
439C2×C4#93811yescharacteristic
440C2×C4#93911yescharacteristic
441C2×C4#94011yescharacteristic
442C8—122no—
443C8—241no—
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—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22#503611yescharacteristic
468C22#503711yescharacteristic
469C22#503811yescharacteristic
470C22#503911yescharacteristic
471C22#521711yescharacteristic
472C22#522311yescharacteristic
473C22#523811yescharacteristic
474C22#524312yes—
475C22#524612yes—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—82no—
484C4—124no—
485C4—242no—
486C3#991911yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—161no—
492C2#7168411yescharacteristic
493C2#7184911yescharacteristic
494C2#7186811yescharacteristic
495C2#7187412yes—
496C2#7188612yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2618C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2618D3
hypocenterC3#9919
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
3. C2×#4981C2semidirect
5. C2×#4528C2semidirect
6. #69501C2semidirect
7. #67624C2semidirect
486. C3#9919semidirect

Maximal subgroups

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

subgroupindex
C3×#26182
C2×#50332
C2×#49812
C2×#45372
C2×#45282
#695012
#676242
#99193