Finite Group Databasebeta

← #10801783

Subgroups — #10801783

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic
2C2×#5101C211yesmaximal, characteristic
3C2×#4853C211yesmaximal, characteristic, complemented
4#73871C212yesmaximal
5C3×#1807C211yesmaximal, characteristic
6#70575C211yesmaximal, characteristic, complemented
7#70561C211yesmaximal, characteristic, complemented
8#70553C211yesmaximal, characteristic, complemented
9#69516C212yesmaximal
10#68850C212yesmaximal, complemented
11#68781C212yesmaximal, complemented
12#21530—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C2×#924C2212yes—
16C22×#208C2211yescharacteristic
17C22×Dic12C2211yescharacteristic
18C2×#887C2212yes—
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C3×#380C2212yes—
22#5116C418yescomplemented
23C2×#818C2211yescharacteristic
24C2×#816C2212yes—
25#5101—24no—
26C2×#815C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30Dic3×C4⋊C4C2212yes—
31#4973C2212yes—
32#4972C2212yes—
33C2×#810C2212yes—
34C2×#810C2212yes—
35#4945—22no—
36#4944—22no—
37#4853—24no—
38#4655—22no—
39#4647—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#467—31no—
44C2×#441—31no—
45C2×#405—31no—
46#2323—32no—
47#1971—31no—
48#1943—31no—
49#1933—32no—
50#1894—32no—
51#1807D311yescharacteristic
52#1776—31no—
53C4×#103—32no—
54C23×C3×C4C2311yescharacteristic
55C22×Dic6C2312yes—
56C22×C3×C8—22no—
57C22×C3×C8—22no—
58C22×C3×C8C2×C414yes—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—22no—
64C2×#217—22no—
65C2×#217D412yes—
66#926—22no—
67#926—22no—
68C2×C4⋊Dic3D412yes—
69C2×C4⋊Dic3C2312yes—
70#924—24no—
71#924C2×C418yes—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2311yescharacteristic
78C2×C4×Dic3C2311yescharacteristic
79C2×#208—22no—
80C2×#208—22no—
81C2×#208—22no—
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C2×Dic12—22no—
85C2×Dic12—22no—
86C2×Dic12C2×C414yes—
87#887—24no—
88#887—24no—
89C3×#106—22no—
90C3×#106—22no—
91C3×#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96#818C2×C414yes—
97#816C2×C418yes—
98#815—22no—
99#815D414yes—
100C8×Dic3—22no—
101C8×Dic3—22no—
102#810—24no—
103#810—24no—
104C22×Q16—31no—
105C22×M16—31no—
106C23×C8C2×D311yescharacteristic
107C2×C22⋊Q8—32no—
108C2×C4×Q8—32no—
109C2×#117—31no—
110#467—38no—
111#441—64no—
112C2×#107—31no—
113C2×#106C2×D311yescharacteristic
114C2×#106C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118#405—64no—
119C2×#97—31no—
120C2×#97—31no—
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#391—32no—
124#389—32no—
125#389—62no—
126#388—62no—
127#388—62no—
128#387—62no—
129#380C2×D312yes—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134C24×C3C2×D411yescharacteristic
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4112yes—
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3C2×D412yes—
148C22×Dic3C2×D412yes—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6C22×C414yes—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—28no—
158C2×C3×C8C22×C412yes—
159C2×C3×C8C22×C412yes—
160C2×C3×C8C2×D412yes—
161C2×C3×C8#4112yes—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C22⋊C4—82no—
167#217—28no—
168#217—24no—
169C4⋊Dic3—22no—
170C4⋊Dic3—24no—
171C4⋊Dic3C22×C414yes—
172Dic3⋊C4—28no—
173Dic3⋊C4—44no—
174Dic3⋊C4—82no—
175C4×Dic3—21no—
176C4×Dic3—21no—
177C4×Dic3—44no—
178C4×Dic3—82no—
179C4×Dic3C2×D412yes—
180C4×Dic3#4112yes—
181#208—28no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8C2×D412yes—
187C2×C3⋊C8#4112yes—
188Dic12—28no—
189C22×Q8—32no—
190C23×C4C22×D311yescharacteristic
191C2×Q16—34no—
192C2×Q16—62no—
193C2×Q16—62no—
194C2×M16—62no—
195C2×M16—62no—
196C2×M16—62no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—31no—
200C22×C8—31no—
201C22×C8C4×D314yes—
202C22×C8C22×D311yescharacteristic
203C22×C8C22×D311yescharacteristic
204C22⋊Q8—38no—
205C22⋊Q8—64no—
206C4×Q8—64no—
207C4×Q8—64no—
208#117—62no—
209#117—62no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4C22×D312yes—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—32no—
219C2×C22⋊C4—62no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—62no—
223C2×C42—62no—
224#107—34no—
225#106—22no—
226#106—22no—
227#103—38no—
228#103—64no—
229#103—64no—
230#97—34no—
231#97—62no—
232C4×C8—62no—
233C4×C8—62no—
234#95—22no—
235#95—62no—
236#95—62no—
237#95—62no—
238#95—62no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3C4×D412yes—
244C23×C3C22×D411yescharacteristic
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4C4×D412yes—
258C2×C3×C4C23×C411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3#12114yes—
276Dic6—24no—
277Dic6—24no—
278Dic6—42no—
279Dic6—42no—
280C3×C8—22no—
281C3×C8C4×D414yes—
282C3⋊C8—22no—
283C3⋊C8—22no—
284C24D3×D411yescharacteristic
285C2×Q8—34no—
286C2×Q8—62no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—62no—
290C2×Q8—62no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4C2×C4×D312yes—
302C22×C4D3×D411yescharacteristic
303C22×C4#23912yes—
304C22×C4C23×D311yescharacteristic
305Q16—68no—
306M16—68no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—28no—
310C2×C8—32no—
311C2×C8—32no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8C2×C4×D312yes—
317C2×C8C2×C4×D312yes—
318C2×C8D3×D412yes—
319C2×C8#23712yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—34no—
325C4⋊C4—64no—
326C4⋊C4—68no—
327C4⋊C4—62no—
328C4⋊C4—124no—
329C4⋊C4—242no—
330C22⋊C4—64no—
331C22⋊C4—68no—
332C22⋊C4—82no—
333C42—32no—
334C42—32no—
335C42—61no—
336C42—61no—
337C42—124no—
338C42—242no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3C2×C4×D411yescharacteristic
345C22×C3C2×#12111yescharacteristic
346C22×C3#53211yescharacteristic
347C22×C3#57712yes—
348C22×C3#57812yes—
349C3×C4—22no—
350C3×C4—82no—
351C3×C4—82no—
352C3×C4C2×C4×D411yescharacteristic
353C3×C4C4×#4111yescharacteristic
354C3×C4#52511yescharacteristic
355C3×C4#52911yescharacteristic
356Dic3—24no—
357Dic3—44no—
358Dic3—42no—
359Dic3—44no—
360C23—22no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23#89312yes—
365C23C2×D3×D411yescharacteristic
366Q8—64no—
367Q8—64no—
368Q8—122no—
369Q8—122no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—34no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—64no—
386C2×C4—64no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4#89312yes—
395C2×C4C22×C4×D311yescharacteristic
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×#23711yescharacteristic
398C2×C4C2×#23911yescharacteristic
399C2×C4D3×#4111yescharacteristic
400C2×C4D3×#4111yescharacteristic
401C8—22no—
402C8—62no—
403C8—62no—
404C8#67⋊C414yes—
405C2×C3—42no—
406C2×C3C4×#12111yescharacteristic
407C2×C3#295011yescharacteristic
408C2×C3#295211yescharacteristic
409C2×C3#306312yes—
410C2×C3#317812yes—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—42no—
415C22—42no—
416C22C2×#89311yescharacteristic
417C22#559611yescharacteristic
418C22D3×#12111yescharacteristic
419C22#576212yes—
420C22#577112yes—
421C4—22no—
422C4—64no—
423C4—82no—
424C4—82no—
425C4—124no—
426C4—122no—
427C4—124no—
428C4C2×#67⋊C411yescharacteristic
429C4#552211yescharacteristic
430C4#562411yescharacteristic
431C4#562711yescharacteristic
432C3#2153011yescharacteristic, complemented, Sylow
433C2—42no—
434C2#7386511yescharacteristic
435C2#7910611yescharacteristic
436C2#7916911yescharacteristic
437C2#8063612yes—
438C2#8066212yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1807C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenter#1807D3
hypocenterC3#21530
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
3. C2×#4853C2semidirect
6. #70575C2semidirect
7. #70561C2semidirect
8. #70553C2semidirect
10. #68850C2semidirect
11. #68781C2semidirect
22. #5116C4semidirect
432. C3#21530semidirect

Maximal subgroups

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

subgroupindex
C2×#51162
C2×#51012
C2×#48532
#738712
C3×#18072
#705752
#705612
#705532
#695162
#688502
#687812
#215303