Finite Group Databasebeta

← #10814412

Subgroups — #10814412

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82338C211yesmaximal, characteristic
2C2×C4×#214C211yesmaximal, characteristic
3#74157C211yesmaximal, characteristic, complemented
4C3×#1804C211yesmaximal, characteristic
5#70486C211yesmaximal, characteristic, complemented
6#70466C211yesmaximal, characteristic, complemented
7#70213C212yesmaximal
8#70188C212yesmaximal
9#69547C211yesmaximal, characteristic, complemented
10#68789C212yesmaximal, complemented
11#68783C212yesmaximal, complemented
12#22765—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C22×#214C2211yescharacteristic
15C2×#887C2212yes—
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102C2212yes—
19C3×#382C2211yescharacteristic
20#5221C2212yes—
21C2×#832C2211yescharacteristic
22#5211—21no—
23C4×#217—21no—
24C2×#816C2212yes—
25C2×#815C2212yes—
26C4×#214C418yescomplemented
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#810C2211yescharacteristic
30C2×#808C2211yescharacteristic
31#4951—22no—
32#4936C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34C2×C4×C3⋊C8C2211yescharacteristic
35#4650—21no—
36#4649—21no—
37#4647—22no—
38C3×#340C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2357—31no—
45#2351—31no—
46#1936—32no—
47#1920—32no—
48#1880—32no—
49#1835—31no—
50#1805—31no—
51#1804D311yescharacteristic
52#1770—31no—
53#1763—32no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×Dic6C2311yescharacteristic
57C22×C3×C8C2312yes—
58C3×C4×D4—22no—
59C3×C4×D4C2×C414yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#217—22no—
65C2×#217—22no—
66C2×#214—22no—
67C2×#214—22no—
68C2×#214C2×C414yes—
69C2×C4⋊Dic3D412yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2312yes—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81#887—24no—
82#887—42no—
83C4×Dic6—22no—
84C4×Dic6C2×C414yes—
85C3×C4⋊C8—22no—
86C3×#102—42no—
87C3×#95—21no—
88C3×#95—21no—
89#832C2×C414yes—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95D411yescharacteristic
95C3⋊#95D411yescharacteristic
96#816—24no—
97#815—42no—
98#810C2×C414yes—
99#808C2×C414yes—
100C4×C3⋊C8C2×C414yes—
101C22×SD16—31no—
102C2×C4×Q8—32no—
103C2×C4×Q8—31no—
104C2×C4×D4C2×D311yescharacteristic
105C4×SD16—38no—
106C2×#106—31no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#103—31no—
109C2×#103—32no—
110C2×#102—31no—
111C2×#102C2×D312yes—
112C2×#97—32no—
113C2×C4×C8—31no—
114#390—32no—
115#389—32no—
116#389—62no—
117#388—62no—
118#382—31no—
119#382—61no—
120#382C2×D311yescharacteristic
121#381—61no—
122C4×C4⋊C4—31no—
123C4×C4⋊C4—32no—
124C4×C4⋊C4—61no—
125C4×C22⋊C4—61no—
126#340—32no—
127#340—32no—
128#340—32no—
129#340C2×D312yes—
130#336—32no—
131C24×C3—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4C22×C412yes—
138C22×C3×C4—21no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—82no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3C2×D412yes—
149C22×Dic3C2×D412yes—
150C2×Dic6—21no—
151C2×Dic6—21no—
152C2×Dic6—21no—
153C2×Dic6C22×C412yes—
154C2×Dic6#4112yes—
155C2×Dic6#4112yes—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4C22×C412yes—
163C3×C22⋊C4—24no—
164C3×C42C22×C412yes—
165C3×C42#4112yes—
166#217—44no—
167#217—82no—
168#214—28no—
169C4⋊Dic3—21no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—42no—
173C4⋊Dic3C22×C412yes—
174Dic3⋊C4—24no—
175Dic3⋊C4—44no—
176Dic3⋊C4—44no—
177C4×Dic3—24no—
178C4×Dic3—22no—
179C4×Dic3—22no—
180C4×Dic3—44no—
181C4×Dic3—82no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8C22×C412yes—
187C2×C3⋊C8C22×C412yes—
188C22×Q8—31no—
189C22×D4C22×D311yescharacteristic
190C23×C4—21no—
191C2×SD16—34no—
192C2×SD16—62no—
193C2×SD16—62no—
194C22×C8—31no—
195C22×C8—31no—
196C22×C8C22×D312yes—
197C4×Q8—34no—
198C4×Q8—62no—
199C4×Q8—64no—
200C4×Q8—122no—
201C4×D4—22no—
202C4×D4C4×D314yes—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—32no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—62no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4C22×D312yes—
210C2×C4⋊C4C22×D311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C42—32no—
215C2×C42—61no—
216C2×C42—62no—
217C2×C42—62no—
218C2×C42C22×D311yescharacteristic
219#106—34no—
220C4⋊C8—22no—
221#103—34no—
222#103—64no—
223#102—34no—
224#102—42no—
225#97—122no—
226C4×C8—34no—
227#95—21no—
228#95—21no—
229#95—31no—
230#95—31no—
231#95—62no—
232#95—62no—
233#95—61no—
234#95—61no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3C22×D411yescharacteristic
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—41no—
244C3×D4—41no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—84no—
253C2×C3×C4C4×D412yes—
254C2×C3×C4C4×D412yes—
255C2×C3×C4C23×C411yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4H(5,𝔽2)11yescharacteristic
260C2×C3×C42−1+411yescharacteristic
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275Dic6—22no—
276Dic6—22no—
277Dic6—22no—
278Dic6#11714yes—
279C3×C8—42no—
280C3⋊C8—22no—
281C3⋊C8—22no—
282C24—21no—
283C2×Q8—32no—
284C2×Q8—32no—
285C2×Q8—32no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C2×C4×D312yes—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—62no—
303C22×C4—82no—
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4C23×D311yescharacteristic
307SD16—68no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—32no—
312C2×C8—32no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—32no—
321C4⋊C4—64no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—122no—
326C4⋊C4—124no—
327C4⋊C4—124no—
328C4⋊C4C2×C4×D312yes—
329C22⋊C4—24no—
330C22⋊C4—124no—
331C22⋊C4—242no—
332C42—64no—
333C42—62no—
334C42—62no—
335C42—124no—
336C42—242no—
337C42C2×C4×D312yes—
338C42#23712yes—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3C2×C4×D411yescharacteristic
345C22×C3#54611yescharacteristic
346C22×C3#54811yescharacteristic
347C22×C3#57512yes—
348C22×C3#57712yes—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4C2×#11711yescharacteristic
354C3×C4#52011yescharacteristic
355C3×C4#55211yescharacteristic
356C3×C4#55311yescharacteristic
357Dic3—24no—
358Dic3—42no—
359Dic3—42no—
360Dic3—42no—
361Dic3—82no—
362C23—22no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23C2×D3×D411yescharacteristic
368Q8—34no—
369Q8—62no—
370Q8—62no—
371Q8—62no—
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—84no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4#67⋊C412yes—
399C2×C4#67⋊C412yes—
400C2×C4C22×C4×D311yescharacteristic
401C2×C4C2×#23711yescharacteristic
402C2×C4#100411yescharacteristic
403C2×C4#100711yescharacteristic
404C2×C4D3×#4111yescharacteristic
405C2×C4D3×#4111yescharacteristic
406C8—42no—
407C8—62no—
408C8—62no—
409C2×C3—82no—
410C2×C3#235211yescharacteristic
411C2×C3#294311yescharacteristic
412C2×C3#294911yescharacteristic
413C2×C3#332612yes—
414C2×C3#337112yes—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—82no—
419C22—82no—
420C22C2×#67⋊C411yescharacteristic
421C22#561211yescharacteristic
422C22#563311yescharacteristic
423C22#576812yes—
424C22#577512yes—
425C4—22no—
426C4—22no—
427C4—42no—
428C4—64no—
429C4—82no—
430C4—122no—
431C4—122no—
432C4—122no—
433C4—242no—
434C4#550411yescharacteristic
435C4D3×#11711yescharacteristic
436C4#561411yescharacteristic
437C4#563911yescharacteristic
438C3#2276511yescharacteristic, complemented, Sylow
439C2—82no—
440C2#7401211yescharacteristic
441C2#7881911yescharacteristic
442C2#7891611yescharacteristic
443C2#8056612yes—
444C2#8120312yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1804D3
hypocenterC3#22765
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
3. #74157C2semidirect
5. #70486C2semidirect
6. #70466C2semidirect
9. #69547C2semidirect
10. #68789C2semidirect
11. #68783C2semidirect
26. C4×#214C4semidirect
438. C3#22765semidirect

Maximal subgroups

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

subgroupindex
#823382
C2×C4×#2142
#741572
C3×#18042
#704862
#704662
#702132
#701882
#695472
#687892
#687832
#227653