Finite Group Databasebeta

← #10877289

Subgroups — #10877289

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4977C211yesmaximal, characteristic
2#73866C211yesmaximal, characteristic
3C3×#1853C211yesmaximal, characteristic
4#71559C211yesmaximal, characteristic
5#70591C212yesmaximal, complemented
6#70564C212yesmaximal, complemented
7#69930C211yesmaximal, characteristic
8#69908C211yesmaximal, characteristic
9#69504C211yesmaximal, characteristic
10#68905C212yesmaximal, complemented
11#68844C212yesmaximal, complemented
12#27686—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C2×#924C2211yescharacteristic
16C22×C4×Dic3C2211yescharacteristic
17C2×#887C2211yescharacteristic
18C2×C3×#106C2212yes—
19C2×C3×C22⋊C8C2211yescharacteristic
20C3×#380C2211yescharacteristic
21C2×C3⋊#95—21no—
22C2×C3⋊#95—21no—
23C2×#830C2211yescharacteristic
24C2×#816C2212yes—
25C2×Dic3⋊C8C2212yes—
26#4977—21no—
27#4977—21no—
28#4977C2212yes—
29#4977C2212yes—
30Dic3×C4⋊C4C2211yescharacteristic
31#4973C2211yescharacteristic
32#4972C2211yescharacteristic
33C2×#810C2212yes—
34C2×#807C2212yes—
35#4946—21no—
36Dic3×C22⋊C4—21no—
37#4652—21no—
38#4649—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×C22⋊Q8—31no—
45C2×#382—31no—
46#2101—32no—
47#2097—32no—
48#1949—32no—
49#1872—31no—
50#1872—31no—
51#1853D311yescharacteristic
52#1852—31no—
53#1805—32no—
54#1797—31no—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—21no—
58C23×Dic3D411yescharacteristic
59C23×Dic3D411yescharacteristic
60C22×Dic6C2311yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—42no—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×#217—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2311yescharacteristic
73#924—22no—
74#924Q814yes—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3—24no—
79C2×C4×Dic3—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C2×C4×Dic3C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86#887—24no—
87C3×#106—42no—
88C3×C22⋊C8—22no—
89C3×#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#830—22no—
99#816—42no—
100Dic3⋊C8—24no—
101#810—42no—
102#807—42no—
103C2×C22⋊Q8—31no—
104C2×C4×Q8—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C42—31no—
108C2×#107—32no—
109C2×#106C2×D312yes—
110C2×C4⋊C8—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8C2×D311yescharacteristic
115#391—31no—
116#389—31no—
117#387—61no—
118#385—61no—
119#382—32no—
120#382—32no—
121#382—61no—
122#382—61no—
123#380C2×D311yescharacteristic
124C4×C4⋊C4—31no—
125C4×C4⋊C4—61no—
126C4×C22⋊C4—61no—
127C2×#95—61no—
128C2×#95—61no—
129#340—32no—
130#340—32no—
131#340—32no—
132#340C2×D312yes—
133#336—32no—
134C24×C3C2×D411yescharacteristic
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×Q812yes—
145C22×C3×C4#4112yes—
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—42no—
157C22×Dic3—42no—
158C22×Dic3—42no—
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—21no—
166C2×Dic6C2×Q812yes—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—42no—
176#217—24no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—42no—
181C4⋊Dic3C2×Q812yes—
182Dic3⋊C4—24no—
183Dic3⋊C4—42no—
184Dic3⋊C4—42no—
185C4×Dic3—22no—
186C4×Dic3—22no—
187C4×Dic3—22no—
188C4×Dic3—42no—
189C4×Dic3—42no—
190C4×Dic3#4114yes—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C22×Q8—31no—
195C23×C4—21no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—61no—
199C23×C4C22×D311yescharacteristic
200C22×C8—32no—
201C22×C8C22×D312yes—
202C22⋊Q8—34no—
203C22⋊Q8—62no—
204C4×Q8—64no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—122no—
214C2×C4⋊C4C22×D312yes—
215C2×C4⋊C4C22×D311yescharacteristic
216C2×C4⋊C4C22×D311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—61no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—31no—
223C2×C42—64no—
224C2×C42—61no—
225C2×C42—61no—
226#107—122no—
227#106—42no—
228C4⋊C8—64no—
229#103—122no—
230#103—122no—
231C22⋊C8—22no—
232C22⋊C8—62no—
233#95—21no—
234#95—62no—
235#95—62no—
236#95—62no—
237#95—62no—
238#95—61no—
239#95—61no—
240#95—61no—
241#95—61no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3C22⋊Q812yes—
247C23×C3#13512yes—
248C23×C3#13712yes—
249C23×C3C22×D411yescharacteristic
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—84no—
264C2×C3×C4C22⋊Q812yes—
265C2×C3×C4C22×Q811yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4112yes—
268C2×C3×C4H(5,𝔽2)12yes—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—84no—
291C2×Dic3#12312yes—
292C2×Dic3#12312yes—
293Dic6—24no—
294Dic6—42no—
295C3×C8—42no—
296C3⋊C8—42no—
297C24D3×D411yescharacteristic
298C2×Q8—32no—
299C2×Q8—61no—
300C2×Q8—62no—
301C2×Q8—62no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—62no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—122no—
324C22×C4—122no—
325C22×C4—122no—
326C22×C4#23512yes—
327C22×C4D3×D411yescharacteristic
328C22×C4D3×Q812yes—
329C22×C4C23×D311yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—32no—
341C4⋊C4—42no—
342C4⋊C4—64no—
343C4⋊C4—61no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4—122no—
349C22⋊C4—42no—
350C22⋊C4—64no—
351C42—34no—
352C42—62no—
353C42—62no—
354C42—62no—
355C42—122no—
356C42—122no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3#47914yes—
364C22×C3C2×C22⋊Q811yescharacteristic
365C22×C3C2×#12311yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3C2×#13511yescharacteristic
368C22×C3C2×#13711yescharacteristic
369C22×C3#57511yescharacteristic
370C22×C3#57511yescharacteristic
371C3×C4—22no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—82no—
375C3×C4#55312yes—
376C3×C4#55412yes—
377Dic3—22no—
378Dic3—22no—
379Dic3—42no—
380Dic3—41no—
381Dic3—41no—
382Dic3—42no—
383Dic3—82no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#89612yes—
389C23#91512yes—
390C23#91612yes—
391C23C2×D3×D411yescharacteristic
392Q8—64no—
393Q8—122no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—84no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—244no—
432C2×C4#89612yes—
433C2×C4C2×#23511yescharacteristic
434C2×C4#100412yes—
435C2×C4C2×D3×Q811yescharacteristic
436C2×C4D3×#4112yes—
437C8—42no—
438C8—122no—
439C2×C3—24no—
440C2×C3#248711yescharacteristic
441C2×C3C2×#47911yescharacteristic
442C2×C3#308011yescharacteristic
443C2×C3#309111yescharacteristic
444C2×C3#309511yescharacteristic
445C2×C3#321111yescharacteristic
446C2×C3#321511yescharacteristic
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22#485714yes—
454C22C2×#89611yescharacteristic
455C22#559211yescharacteristic
456C22D3×#12311yescharacteristic
457C22C2×#91511yescharacteristic
458C22C2×#91611yescharacteristic
459C22#577311yescharacteristic
460C22#577711yescharacteristic
461C4—22no—
462C4—62no—
463C4—62no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—121no—
468C4—122no—
469C4—121no—
470C4—122no—
471C4—242no—
472C4#554012yes—
473C4#563612yes—
474C3#2768611yescharacteristic, complemented, Sylow
475C2—24no—
476C2#7445711yescharacteristic
477C2C2×#485711yescharacteristic
478C2#7924111yescharacteristic
479C2#8005711yescharacteristic
480C2#8014011yescharacteristic
481C2#8132411yescharacteristic
482C2#8142811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1853C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1853D3
hypocenterC3#27686
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
5. #70591C2semidirect
6. #70564C2semidirect
10. #68905C2semidirect
11. #68844C2semidirect
474. C3#27686semidirect

Maximal subgroups

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

subgroupindex
C2×#49772
#738662
C3×#18532
#715592
#705912
#705642
#699302
#699082
#695042
#689052
#688442
#276863