Finite Group Databasebeta

← #10690360

Subgroups — #10690360

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4722C211yesmaximal, characteristic
2#73874C211yesmaximal, characteristic
3#69502C211yesmaximal, characteristic
4#69422C211yesmaximal, characteristic
5C2×#4452C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#9046—31nomaximal, Sylow
9C22×C4⋊Dic3C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14C2×C3⋊#95—21no—
15C2×C3⋊#95—21no—
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#818—21no—
19C2×#816—21no—
20#4978—21no—
21#4976—21no—
22#4946—21no—
23Dic3×C22⋊C4—21no—
24#4722—22no—
25#4722C414yescomplemented
26#4653—21no—
27#4646—21no—
28#4529—21no—
29#4529—21no—
30#4452—22no—
31C2×#484—31no—
32#2329—31no—
33#1824—31no—
34C2×#328—31no—
35#1445—31no—
36#1294D311yescharacteristic
37#1280—31no—
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C23×Dic3D411yescharacteristic
42C23×Dic3D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2×C412yes—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58#924C2×C412yes—
59#924D412yes—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—21no—
65C22×C3⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8C2×C412yes—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—42no—
78C3⋊#95—42no—
79#830—21no—
80#830—21no—
81#818—24no—
82#816—24no—
83C2×C22⋊Q8—31no—
84C22×C4⋊C4—31no—
85#484—34no—
86#484—62no—
87C2×#107—61no—
88C2×#103—61no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8C2×D311yescharacteristic
91#391—61no—
92#389—61no—
93#387—61no—
94#387—61no—
95#385—61no—
96#380C2×D311yescharacteristic
97C4×C22⋊C4—61no—
98C2×#95—31no—
99C2×#95—61no—
100C2×#95—61no—
101C2×#95C2×D311yescharacteristic
102#340—61no—
103#340—61no—
104#328—62no—
105C24×C3C2×D411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3Q1612yes—
142C2×Dic6—21no—
143C2×Dic6—41no—
144C2×Dic6—41no—
145C2×Dic6C22⋊C412yes—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—22no—
154#217—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3C22⋊C412yes—
161Dic3⋊C4—22no—
162Dic3⋊C4—22no—
163Dic3⋊C4—81no—
164Dic3⋊C4—81no—
165C4×Dic3—81no—
166C4×Dic3—81no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C22×Q8—31no—
171C23×C4—21no—
172C23×C4—31no—
173C23×C4—31no—
174C23×C4—61no—
175C23×C4C22×D311yescharacteristic
176C22×C8—21no—
177C22×C8—61no—
178C22⋊Q8—32no—
179C22⋊Q8—32no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4D1211yescharacteristic
193C2×C4⋊C4D1211yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—61no—
196C2×C42—61no—
197C2×C42—61no—
198#107—64no—
199#103—64no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8C4×D312yes—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—122no—
214#95—122no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3C4×D411yescharacteristic
220C23×C3#12011yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3C2×Q1611yescharacteristic
223C23×C3#13511yescharacteristic
224C23×C3#13711yescharacteristic
225C23×C3#13711yescharacteristic
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—81no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274Dic6—42no—
275Dic6—81no—
276C3×C8—42no—
277C3⋊C8—81no—
278C24D3×D411yescharacteristic
279C2×Q8—32no—
280C2×Q8—61no—
281C2×Q8—121no—
282C2×Q8—121no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—32no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—62no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4C2×D1211yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4#23911yescharacteristic
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—121no—
323C2×C8—121no—
324C2×C8—121no—
325C4⋊C4—32no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—62no—
329C4⋊C4—61no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C4⋊C4—241no—
336C4⋊C4—241no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—62no—
340C22⋊C4—62no—
341C42—241no—
342C42—241no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3#38611yescharacteristic
356C22×C3C4×Q1611yescharacteristic
357C22×C3#44111yescharacteristic
358C22×C3#45111yescharacteristic
359C22×C3#45211yescharacteristic
360C22×C3#45311yescharacteristic
361C22×C3#45411yescharacteristic
362C22×C3#45811yescharacteristic
363C22×C3#45811yescharacteristic
364C22×C3#46211yescharacteristic
365C22×C3#46411yescharacteristic
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4—81no—
373Dic3—42no—
374Dic3—42no—
375Dic3—42no—
376Dic3—81no—
377Dic3—82no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23#88211yescharacteristic
383C23#88411yescharacteristic
384C23#89311yescharacteristic
385C23#91211yescharacteristic
386C23#91511yescharacteristic
387C23#91511yescharacteristic
388C23D3×Q1611yescharacteristic
389Q8—122no—
390Q8—241no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4—241no—
432C2×C4—241no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4C4×D1211yescharacteristic
436C2×C4D3×C22⋊C411yescharacteristic
437C2×C4#89411yescharacteristic
438C2×C4#89511yescharacteristic
439C8—42no—
440C8—241no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#161411yescharacteristic
446C2×C3#161511yescharacteristic
447C2×C3#161711yescharacteristic
448C2×C3#161811yescharacteristic
449C2×C3#187811yescharacteristic
450C2×C3#188311yescharacteristic
451C2×C3#189011yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22#466511yescharacteristic
465C22#474211yescharacteristic
466C22#474211yescharacteristic
467C22#476711yescharacteristic
468C22#478311yescharacteristic
469C22#479511yescharacteristic
470C22#480911yescharacteristic
471C22#482211yescharacteristic
472C22#482711yescharacteristic
473C22#485311yescharacteristic
474C22#487311yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—82no—
480C4—81no—
481C4—81no—
482C4—122no—
483C4—122no—
484C4—122no—
485C4—242no—
486C4—241no—
487C3#904611yescharacteristic, complemented, Sylow
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2#6829111yescharacteristic
493C2#6829811yescharacteristic
494C2#6832211yescharacteristic
495C2#6833111yescharacteristic
496C2#6877611yescharacteristic
497C2#6890811yescharacteristic
498C2#6896311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9046
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
25. #4722C4semidirect
487. C3#9046semidirect

Maximal subgroups

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

subgroupindex
C2×#47222
#738742
#695022
#694222
C2×#44522
C3×#12942
#676742
#90463