Finite Group Databasebeta

← #10692508

Subgroups — #10692508

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2543C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#71675C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic, complemented
5#67612C212yesmaximal, complemented
6#9384—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×#121C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#398C411yescharacteristic, complemented
11C3×#397C411yescharacteristic, complemented
12C3×#394C411yescharacteristic, complemented
13C3×#392C411yescharacteristic, complemented
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C2×C3×#95C2211yescharacteristic
18C2×#834—21no—
19#5212C2212yes—
20C2×#830—21no—
21C2×#830C2212yes—
22D4×C3⋊C8—24no—
23#4925—21no—
24#4537—24no—
25#4526—21no—
26#2972—31no—
27#2543D311yescharacteristic
28#1799—31no—
29#1519—32no—
30#1294—32no—
31C22×C3×D4—21no—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—42no—
41C3×#121—42no—
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×#217—22no—
56C2×C4⋊Dic3—22no—
57C22×C3⋊C8—22no—
58C22×C3⋊C8—22no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3×#95C2×C411yescharacteristic
64#834—42no—
65C3⋊#95—22no—
66#830—22no—
67#830—22no—
68#830D414yescomplemented
69C4⋊(C3⋊C8)—22no—
70C4×C3⋊C8—22no—
71C2×#121—21no—
72C2×#121C2×D311yescharacteristic
73C2×C4×D4C2×D311yescharacteristic
74C8×D4—64no—
75#431—61no—
76C2×#99—61no—
77C2×C22⋊C8—32no—
78C2×C22⋊C8—61no—
79#398Dic311yescharacteristic, complemented
80#397Dic311yescharacteristic, complemented
81#394Dic311yescharacteristic, complemented
82#392Dic311yescharacteristic, complemented
83#388—21no—
84#386—21no—
85#386—21no—
86#380—32no—
87C2×#95C2×D311yescharacteristic
88#342—61no—
89#327—64no—
90C24×C3—21no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—42no—
102C2×C3×D4—81no—
103C2×C3×D4—81no—
104C2×C3×D4D812yes—
105C2×C3×D4SD1612yes—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—81no—
115C22×C3×C4—81no—
116C22×C3×C4—81no—
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22⋊C411yescharacteristic
119C22×C3×C4C22⋊C411yescharacteristic
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4112yes—
124C22×Dic3—42no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—42no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—81no—
136C3×C42—21no—
137C3×C42—21no—
138#217—42no—
139#217—42no—
140C4⋊Dic3—82no—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—82no—
143C2×C3⋊C8—82no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C2×Dic311yescharacteristic
147C22×D4#6711yescharacteristic
148C23×C4—21no—
149C23×C4#6711yescharacteristic
150C23×C4C22×D311yescharacteristic
151C22×C8—62no—
152C22×C8—62no—
153#121—22no—
154#121—22no—
155#121—42no—
156#121—42no—
157C4×D4—22no—
158C4×D4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4C2×Dic311yescharacteristic
162C2×C4⋊C4C2×Dic311yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4C2×Dic311yescharacteristic
170C2×C22⋊C4#6711yescharacteristic
171C2×C42#6711yescharacteristic
172C4⋊C8—62no—
173#99—122no—
174C22⋊C8—34no—
175C22⋊C8—62no—
176C22⋊C8—62no—
177C4×C8—62no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—62no—
182#95C2×Dic311yescharacteristic
183#95C2×Dic311yescharacteristic
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3#10212yes—
198C23×C3C2×C22⋊C411yescharacteristic
199C23×C3#12112yes—
200C23×C3C2×D811yescharacteristic
201C23×C3C2×SD1611yescharacteristic
202C23×C3#13512yes—
203C3×D4—24no—
204C3×D4—24no—
205C3×D4—82no—
206C3×D4—84no—
207C3×D4—82no—
208C3×D4—82no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—82no—
231C2×C3×C4—82no—
232C2×C3×C4#10212yes—
233C2×C3×C4C4×D412yes—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×Dic3—82no—
237C2×Dic3—82no—
238C2×Dic3—82no—
239C3⋊C8—44no—
240C24—21no—
241C24#21711yescharacteristic
242C24C2×#6711yescharacteristic
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—81no—
253C2×D4—81no—
254C2×D4#21312yes—
255C2×D4#21412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—122no—
268C22×C4#21711yescharacteristic
269C22×C4#21711yescharacteristic
270C22×C4#21711yescharacteristic
271C22×C4D3×D412yes—
272C22×C4#23712yes—
273C22×C4C22×Dic311yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C2×C8—64no—
276C2×C8—242no—
277C2×C8—242no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—242no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—81no—
290C22⋊C4—122no—
291C22⋊C4—122no—
292C42—21no—
293C42—21no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#38611yescharacteristic
314C22×C3C2×#10211yescharacteristic
315C22×C3#41611yescharacteristic
316C22×C3#46512yes—
317C22×C3#46612yes—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4—81no—
327Dic3—162no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23#83212yes—
342C23C2×#21311yescharacteristic
343C23C2×#21411yescharacteristic
344C23#93912yes—
345C23#95012yes—
346C23C2×#21711yescharacteristic
347D4—24no—
348D4—24no—
349D4—82no—
350D4—84no—
351D4—82no—
352D4—82no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—242no—
377C2×C4—242no—
378C2×C4—242no—
379C2×C4#83212yes—
380C2×C4D4×Dic312yes—
381C2×C4#93511yescharacteristic
382C2×C4#93711yescharacteristic
383C8—124no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—22no—
388C2×C3—42no—
389C2×C3—161no—
390C2×C3#188411yescharacteristic
391C2×C3#189311yescharacteristic
392C2×C3#191311yescharacteristic
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#503812yes—
413C22#504112yes—
414C22C2×#83211yescharacteristic
415C22#522011yescharacteristic
416C22#523311yescharacteristic
417C4—22no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—81no—
425C4—81no—
426C4—482no—
427C3#938411yescharacteristic, complemented, Sylow
428C2—21no—
429C2—22no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—161no—
434C2#7168111yescharacteristic
435C2#7181711yescharacteristic
436C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2543C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2543D3
hypocenterC3#9384
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #71675C2semidirect
4. #71619C2semidirect
5. #67612C2semidirect
10. C3×#398C4semidirect
11. C3×#397C4semidirect
12. C3×#394C4semidirect
13. C3×#392C4semidirect
68. #830D4semidirect
79. #398Dic3semidirect
80. #397Dic3semidirect
81. #394Dic3semidirect
82. #392Dic3semidirect
427. C3#9384semidirect

Maximal subgroups

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

subgroupindex
C3×#25432
#823012
#716752
#716192
#676122
#93843