Finite Group Databasebeta

← #10824697

Subgroups — #10824697

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×Dic3×Q16C211yesmaximal, characteristic, complemented
2#82744C212yesmaximal
3C2×#4720C211yesmaximal, characteristic
4C3×#2001C211yesmaximal, characteristic
5#72068C211yesmaximal, characteristic, complemented
6#71695C212yesmaximal
7#70262C212yesmaximal, complemented
8#70220C212yesmaximal, complemented
9#69938C211yesmaximal, characteristic
10#68838C211yesmaximal, characteristic, complemented
11#68785C211yesmaximal, characteristic, complemented
12#23705—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×Q8×Dic3C2212yes—
16C2×#926C2211yescharacteristic
17C2×C3×#103C2212yes—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5229C2212yes—
20C2×#835C2212yes—
21C2×#835C2212yes—
22#5212C2212yes—
23C2×#830C2211yescharacteristic
24Dic3×Q16—24no—
25C2×#818C2211yescharacteristic
26C2×#815C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4984—22no—
31#4984C2212yes—
32#4980—22no—
33Dic3×C4⋊C4C2212yes—
34C2×#807C2212yes—
35#4951—22no—
36#4951—22no—
37#4720—24no—
38C3×#340C2211yescharacteristic
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#433—31no—
44#2323—32no—
45#2086—31no—
46#2075—31no—
47#2068—32no—
48#2056—31no—
49#2036—32no—
50#2001D311yescharacteristic
51C4×#107—31no—
52#1763—32no—
53C22×C3×Q8C2312yes—
54C23×C3×C4C2311yescharacteristic
55C2×C3×Q16—24no—
56C2×C3×Q16—22no—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—22no—
64C2×#217—22no—
65C2×#217D411yescharacteristic
66C2×#217D411yescharacteristic
67Q8×Dic3—24no—
68Q8×Dic3—24no—
69#926—22no—
70#926—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C3×#103—24no—
84C3×C22⋊C8—22no—
85#835—24no—
86#835—24no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92#830—22no—
93#818—22no—
94#815—22no—
95Dic3⋊C8—22no—
96Dic3⋊C8—22no—
97C8×Dic3D414yes—
98#807—24no—
99C22×Q16C2×D311yescharacteristic
100C2×C22⋊Q8C2×D312yes—
101C2×C4×Q8—32no—
102C2×#117—31no—
103C4×Q16—64no—
104#433—64no—
105C2×#107—31no—
106C2×#107—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103C2×D312yes—
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#380—32no—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340—32no—
127#340C2×D311yescharacteristic
128C24×C3C2×D411yescharacteristic
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8#4114yes—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3C2×D411yescharacteristic
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C3×Q16—28no—
150C2×C3×C8—22no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8C2×D412yes—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4#4114yes—
157C3×C22⋊C4—28no—
158#217—22no—
159#217—22no—
160#217—82no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—44no—
168C4⋊Dic3—82no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—44no—
172C4×Dic3—82no—
173C4×Dic3C2×D412yes—
174C4×Dic3#4112yes—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8C2×D412yes—
179C22×Q8C22×D312yes—
180C23×C4C22×D311yescharacteristic
181C2×Q16—24no—
182C2×Q16—22no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C4×Q8—64no—
190C4×Q8—64no—
191#117—62no—
192#117—62no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—62no—
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—62no—
210#107—64no—
211C4⋊C8—62no—
212C4⋊C8—62no—
213#103—24no—
214#103—64no—
215#103—64no—
216C22⋊C8—22no—
217C22⋊C8—62no—
218#97—62no—
219C4×C8—34no—
220#95—62no—
221#95—62no—
222#95—62no—
223#95—62no—
224#95—62no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3C22×D411yescharacteristic
231C3×Q8—24no—
232C3×Q8—24no—
233C3×Q8—42no—
234C3×Q8—42no—
235C2×C3×C4—21no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4112yes—
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4111yescharacteristic
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—22no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—22no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3#12112yes—
265C2×Dic3#12112yes—
266C3×C8—22no—
267C3×C8—41no—
268C3⋊C8—22no—
269C3⋊C8—41no—
270C24D3×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8#23714yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
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—28no—
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—28no—
301C4⋊C4—22no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—124no—
309C4⋊C4—242no—
310C4⋊C4#23714yes—
311C22⋊C4—28no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—242no—
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#57611yescharacteristic
329C22×C3#57811yescharacteristic
330C22×C3#57812yes—
331C3×C4—44no—
332C3×C4—41no—
333C3×C4—44no—
334C3×C4#52511yescharacteristic
335C3×C4#52912yes—
336C3×C4#53211yescharacteristic
337Dic3—22no—
338Dic3—22no—
339Dic3—41no—
340Dic3—41no—
341Dic3—82no—
342Dic3—82no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23C2×D3×D411yescharacteristic
349Q8—24no—
350Q8—24no—
351Q8—42no—
352Q8—42no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—62no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4C2×D3×D411yescharacteristic
381C2×C4C2×#23712yes—
382C2×C4C2×#23712yes—
383C2×C4D3×#4111yescharacteristic
384C8—22no—
385C8—41no—
386C8—62no—
387C8—121no—
388C2×C3—81no—
389C2×C3#239211yescharacteristic
390C2×C3#305911yescharacteristic
391C2×C3#306111yescharacteristic
392C2×C3#312412yes—
393C2×C3#312712yes—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22#558612yes—
400C22D3×#12111yescharacteristic
401C22#576611yescharacteristic
402C22#576911yescharacteristic
403C22#577912yes—
404C4—44no—
405C4—41no—
406C4—44no—
407C4—62no—
408C4—62no—
409C4—121no—
410C4—121no—
411C4—242no—
412C4—242no—
413C4#558611yescharacteristic
414C4#560012yes—
415C4#571611yescharacteristic
416C3#2370511yescharacteristic, complemented, Sylow
417C2—81no—
418C2#7666611yescharacteristic
419C2#8017712yes—
420C2#8021812yes—
421C2#8076111yescharacteristic
422C2#8090511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2001D3
hypocenterC3#23705
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×Q16C2semidirect
5. #72068C2semidirect
7. #70262C2semidirect
8. #70220C2semidirect
10. #68838C2semidirect
11. #68785C2semidirect
416. C3#23705semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×Q162
#827442
C2×#47202
C3×#20012
#720682
#716952
#702622
#702202
#699382
#688382
#687852
#237053