Finite Group Databasebeta

← #10691012

Subgroups — #10691012

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3109C211yesmaximal, characteristic
2#82303C211yesmaximal, characteristic, complemented
3#71811C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71609C211yesmaximal, characteristic, complemented
6#67751C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#9163—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#481—21no—
12C3×#476C411yescharacteristic, complemented
13C3×#476C411yescharacteristic, complemented
14C3×#459C411yescharacteristic, complemented
15C3×#459C411yescharacteristic, complemented
16C3×#447—21no—
17C3×#442—21no—
18C3×#442—21no—
19C3×#432C2211yescharacteristic
20C2×#832—21no—
21#5207—21no—
22C2×#830—21no—
23#5012—21no—
24#5011—22no—
25D4×C3⋊C8—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4540—22no—
31#4537—21no—
32#4537—21no—
33#4529—21no—
34#3109D311yescharacteristic
35#2974—31no—
36#1762—31no—
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—21no—
45C2×C3×D8—21no—
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#127C2×C411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4C2×C411yescharacteristic
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×C4×D4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—22no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—22no—
67C3×#106—21no—
68C3×#106—21no—
69C3×C4⋊C8C2×C411yescharacteristic
70C3×C4⋊C8C2×C411yescharacteristic
71C3×#102—21no—
72C3×#102—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×C22⋊C8—21no—
76C3×#97—21no—
77#832—24no—
78C3⋊#95—21no—
79#830—22no—
80#830—22no—
81#830—21no—
82#830D411yescharacteristic
83#830D411yescharacteristic
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#532C2×D311yescharacteristic
91C2×C4×D4C2×D311yescharacteristic
92#481—21no—
93#476Dic311yescharacteristic, complemented
94#476Dic311yescharacteristic, complemented
95#459Dic311yescharacteristic, complemented
96#459Dic311yescharacteristic, complemented
97#447—21no—
98#442—21no—
99#442—21no—
100#436—61no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432C2×D311yescharacteristic
105#431—31no—
106C2×#102—61no—
107#407—61no—
108C2×C22⋊C8—61no—
109C2×#97—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#329—62no—
113#327—61no—
114#327—61no—
115C24×C3—21no—
116C3×#41—81no—
117C3×#41—81no—
118C2×C3×Q8—21no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4C22⋊C411yescharacteristic
127C2×C3×D4C22⋊C411yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—82no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—41no—
139C3×D8—42no—
140C3×D8—42no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C22×C411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158C4⋊Dic3—22no—
159C4⋊Dic3—41no—
160C4×Dic3—81no—
161#208—82no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—82no—
169C2×#41—21no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×M16—62no—
175C22×C8—61no—
176C22×C8—62no—
177C4⋊Q8C2×Dic311yescharacteristic
178#127C2×Dic311yescharacteristic
179#124—21no—
180#121—21no—
181#121—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C2×Dic311yescharacteristic
187C4×D4C2×Dic311yescharacteristic
188C4×D4#6711yescharacteristic
189C4×D4#6711yescharacteristic
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—21no—
196#106—21no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8C2×Dic311yescharacteristic
200C4⋊C8C2×Dic311yescharacteristic
201#102—21no—
202#102—21no—
203#102—21no—
204#102—21no—
205#102—64no—
206C22⋊C8—21no—
207C22⋊C8—31no—
208C22⋊C8—31no—
209C22⋊C8—62no—
210C22⋊C8—62no—
211C22⋊C8—61no—
212#97—21no—
213#97—32no—
214#97—61no—
215#97—61no—
216C4×C8—61no—
217#95—61no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—81no—
225C23×C3#12111yescharacteristic
226C3×Q8—81no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—21no—
230C3×D4—42no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4#10412yes—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—82no—
251C2×C3×C4—81no—
252C2×C3×C4C2×C22⋊C411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12311yescharacteristic
258C2×Dic3—42no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3⋊C8—42no—
264C3⋊C8—41no—
265C3⋊C8—42no—
266C3⋊C8—41no—
267C24—21no—
268#41—81no—
269#41—81no—
270C2×Q8—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4#21711yescharacteristic
279C2×D4#21711yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—121no—
288C22×C4D3×D411yescharacteristic
289C22×C4#23711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291D8—42no—
292D8—42no—
293M16—242no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—62no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—62no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—242no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—62no—
307C4⋊C4—121no—
308C4⋊C4#21711yescharacteristic
309C4⋊C4#21711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—241no—
317C42D3×D411yescharacteristic
318C42#23711yescharacteristic
319C42C22×Dic311yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#47111yescharacteristic
336C22×C3#47211yescharacteristic
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—81no—
345C3×C4C2×#10411yescharacteristic
346C3×C4#42111yescharacteristic
347Dic3—82no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—81no—
355C23#93911yescharacteristic
356Q8—81no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4#83712yes—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—122no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4D4×Dic311yescharacteristic
386C2×C4D4×Dic311yescharacteristic
387C2×C4#93511yescharacteristic
388C2×C4#93711yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4C2×#21711yescharacteristic
391C8—41no—
392C8—41no—
393C8—122no—
394C8—121no—
395C8—122no—
396C8—121no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—42no—
400C2×C3—81no—
401C2×C3—161no—
402C2×C3#187111yescharacteristic
403C2×C3#190411yescharacteristic
404C2×C3#190711yescharacteristic
405C22—21no—
406C22—21no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#519911yescharacteristic
419C22#520011yescharacteristic
420C22#522011yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—81no—
429C4—242no—
430C4C2×#83711yescharacteristic
431C4#523611yescharacteristic
432C3#916311yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—42no—
436C2—81no—
437C2—161no—
438C2#7171311yescharacteristic
439C2#7208911yescharacteristic
440C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3109C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3109D3
hypocenterC3#9163
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #82303C2semidirect
3. #71811C2semidirect
4. #71798C2semidirect
5. #71609C2semidirect
6. #67751C2semidirect
7. #67750C2semidirect
12. C3×#476C4semidirect
13. C3×#476C4semidirect
14. C3×#459C4semidirect
15. C3×#459C4semidirect
93. #476Dic3semidirect
94. #476Dic3semidirect
95. #459Dic3semidirect
96. #459Dic3semidirect
432. C3#9163semidirect

Maximal subgroups

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

subgroupindex
C3×#31092
#823032
#718112
#717982
#716092
#677512
#677502
#91633