Finite Group Databasebeta

← #10824688

Subgroups — #10824688

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4873C211yesmaximal, characteristic, complemented
2C2×#4720C211yesmaximal, characteristic
3#73871C212yesmaximal
4C3×#2076C211yesmaximal, characteristic
5#70622C211yesmaximal, characteristic, complemented
6#69964C211yesmaximal, characteristic
7#69526C212yesmaximal
8#68912C212yesmaximal, complemented
9#68864C211yesmaximal, characteristic, complemented
10#68784C211yesmaximal, characteristic, complemented
11#68783C212yesmaximal, complemented
12#23704—31nomaximal, Sylow
13C2×#926C2211yescharacteristic
14C2×#924C2212yes—
15C22×Dic12C2211yescharacteristic
16C2×#887C2212yes—
17C2×C3×#107C2211yescharacteristic
18C2×C3×C22⋊C8C2211yescharacteristic
19C3×#380C2212yes—
20C2×#830C2211yescharacteristic
21C2×#816C2212yes—
22C2×#815C2211yescharacteristic
23C2×Dic3⋊C8C2211yescharacteristic
24C2×Dic3⋊C8C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26Dic3×C4⋊C4C2212yes—
27#4973C2212yes—
28#4972C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#808C2212yes—
32#4945—22no—
33#4944—22no—
34#4873—24no—
35#4720—24no—
36#4655—22no—
37#4647—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#433—31no—
44#2323—32no—
45#2087—31no—
46#2076D311yescharacteristic
47#2066—32no—
48#2056—31no—
49#2035—32no—
50#2001—31no—
51C4×#106—31no—
52#1763—32no—
53C23×C3×C4C2311yescharacteristic
54C22×Dic6C2312yes—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—22no—
60C2×#217—22no—
61C2×#217D411yescharacteristic
62C2×#217D411yescharacteristic
63#926—22no—
64#926—22no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3C2312yes—
68#924—24no—
69#924—24no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2311yescharacteristic
76C2×C4×Dic3C2311yescharacteristic
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79C2×Dic12—24no—
80C2×Dic12—22no—
81#887—24no—
82#887—24no—
83C3×#107—22no—
84C3×C22⋊C8—22no—
85C3×#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—22no—
91#816—24no—
92#815—22no—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95C8×Dic3D414yes—
96#810—24no—
97#810—24no—
98#808—24no—
99C22×Q16—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C2×#117—31no—
103C4×Q16—64no—
104#433—64no—
105C2×#107C2×D311yescharacteristic
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114C2×#97—31no—
115C2×C4×C8—31no—
116#391—32no—
117#389—32no—
118#389—62no—
119#388—62no—
120#388—62no—
121#387—62no—
122#380C2×D312yes—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D312yes—
128C24×C3C2×D411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3C2×D411yescharacteristic
140C22×Dic3C2×D411yescharacteristic
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D411yescharacteristic
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6#4114yes—
149C2×C3×C8—22no—
150C2×C3×C8—21no—
151C2×C3×C8—22no—
152C2×C3×C8C2×D412yes—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C22⋊C4—82no—
158#217—28no—
159#217—22no—
160#217—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3#4114yes—
165Dic3⋊C4—28no—
166Dic3⋊C4—44no—
167Dic3⋊C4—82no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—44no—
171C4×Dic3—82no—
172C4×Dic3C2×D412yes—
173C4×Dic3#4112yes—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8C2×D412yes—
178Dic12—28no—
179C22×Q8—32no—
180C23×C4C22×D311yescharacteristic
181C2×Q16—64no—
182C2×Q16—62no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187C22⋊Q8—64no—
188C22⋊Q8—64no—
189C4×Q8—64no—
190C4×Q8—64no—
191#117—62no—
192#117—62no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4C22×D312yes—
200C2×C4⋊C4C22×D312yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—31no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—62no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—62no—
208C2×C42—62no—
209#107—22no—
210#106—64no—
211C4⋊C8—62no—
212C4⋊C8—62no—
213#103—64no—
214#103—64no—
215#103—64no—
216C22⋊C8—22no—
217C22⋊C8—62no—
218#97—62no—
219C4×C8—34no—
220#95—22no—
221#95—62no—
222#95—62no—
223#95—62no—
224#95—62no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3C22×D411yescharacteristic
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4112yes—
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×Dic3—22no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—22no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3#12112yes—
261C2×Dic3#12112yes—
262Dic6—24no—
263Dic6—24no—
264Dic6—42no—
265Dic6—42no—
266C3×C8—22no—
267C3×C8—41no—
268C3⋊C8—22no—
269C3⋊C8—41no—
270C24D3×D411yescharacteristic
271C2×Q8—34no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×Q8—62no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4D3×D411yescharacteristic
290C22×C4C23×D311yescharacteristic
291Q16—68no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—32no—
296C2×C8—62no—
297C2×C8—61no—
298C2×C8—62no—
299C2×C8D3×D412yes—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—34no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—68no—
308C4⋊C4—62no—
309C4⋊C4—124no—
310C4⋊C4—242no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—68no—
314C22⋊C4—82no—
315C42—32no—
316C42—32no—
317C42—61no—
318C42—61no—
319C42—124no—
320C42—242no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C2×#12111yescharacteristic
327C22×C3#52512yes—
328C22×C3#57712yes—
329C22×C3#57811yescharacteristic
330C22×C3#57811yescharacteristic
331C3×C4—41no—
332C3×C4—82no—
333C3×C4—82no—
334C3×C4#52511yescharacteristic
335C3×C4#52912yes—
336C3×C4#53211yescharacteristic
337Dic3—22no—
338Dic3—22no—
339Dic3—44no—
340Dic3—44no—
341Dic3—41no—
342Dic3—41no—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—21no—
348C23C2×D3×D411yescharacteristic
349Q8—64no—
350Q8—64no—
351Q8—122no—
352Q8—122no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—62no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4C2×D3×D411yescharacteristic
381C2×C4D3×#4112yes—
382C2×C4D3×#4112yes—
383C2×C4D3×#4111yescharacteristic
384C8—22no—
385C8—41no—
386C8—62no—
387C8—121no—
388C2×C3—81no—
389C2×C3#239211yescharacteristic
390C2×C3#305911yescharacteristic
391C2×C3#306011yescharacteristic
392C2×C3#312012yes—
393C2×C3#312112yes—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22D3×#12111yescharacteristic
400C22#565612yes—
401C22#576611yescharacteristic
402C22#577111yescharacteristic
403C22#577512yes—
404C4—41no—
405C4—62no—
406C4—62no—
407C4—82no—
408C4—82no—
409C4—124no—
410C4—121no—
411C4—124no—
412C4—121no—
413C4#565611yescharacteristic
414C4#566811yescharacteristic
415C4#570612yes—
416C3#2370411yescharacteristic, complemented, Sylow
417C2—81no—
418C2#7745711yescharacteristic
419C2#8076011yescharacteristic
420C2#8086611yescharacteristic
421C2#8135312yes—
422C2#8145612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2076C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2076D3
hypocenterC3#23704
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2076

Complements

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

NH ≅ G/Ntype
1. C2×#4873C2semidirect
5. #70622C2semidirect
8. #68912C2semidirect
9. #68864C2semidirect
10. #68784C2semidirect
11. #68783C2semidirect
416. C3#23704semidirect

Maximal subgroups

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

subgroupindex
C2×#48732
C2×#47202
#738712
C3×#20762
#706222
#699642
#695262
#689122
#688642
#687842
#687832
#237043