Finite Group Databasebeta

← #10691916

Subgroups — #10691916

Order 768.

632 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
1C3×#3205C211yesmaximal, characteristic
2#83172C211yesmaximal, characteristic, complemented
3#71816C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#67738C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9299—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#493C412yescomplemented
13C3×#486C412yescomplemented
14C3×#459—22no—
15C3×#447—22no—
16C3×#431C2211yescharacteristic
17C2×#837—21no—
18C2×#837—21no—
19#5221—21no—
20#5220—22no—
21#5220—22no—
22#4936C2211yescharacteristic
23#4932C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4544—22no—
27#4537—22no—
28#3205D311yescharacteristic
29#2328—31no—
30#1963—31no—
31#1545—31no—
32#1517—31no—
33#1286—31no—
34#1274—31no—
35C2×C3×#41—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4D412yes—
39C23×Dic3—22no—
40C2×C3×D8—24no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#127—21no—
43C3×#127—21no—
44C3×#127C2×C412yes—
45C3×#127C2×C411yescharacteristic, complemented
46C3×#127D412yes—
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56D4×Dic3—42no—
57D4×Dic3—42no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62C3×C4⋊C8—21no—
63C3×#102—24no—
64C3×C22⋊C8—21no—
65C3×C4×C8C2×C412yescomplemented
66#837—42no—
67#837—42no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95D412yes—
72#830—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4×C3⋊C8—21no—
77#532C2×D311yescharacteristic
78C2×#127C2×D311yescharacteristic
79C2×C4×D4—61no—
80#493Dic312yescomplemented
81#486Dic312yescomplemented
82#459—22no—
83#447—22no—
84#431—31no—
85#431C2×D311yescharacteristic
86#423—31no—
87C2×#104—61no—
88C2×#104—61no—
89#390—61no—
90#386—62no—
91#386—62no—
92#382—31no—
93#376—31no—
94#331—62no—
95#327—62no—
96C24×C3—22no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—22no—
104C2×C3×D4—21no—
105C2×C3×D4—22no—
106C2×C3×D4—21no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—82no—
116C2×C3×D4—82no—
117C2×C3×D4C22⋊C414yes—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—44no—
126C22×Dic3—82no—
127C22×Dic3—82no—
128C3×D8—84no—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138#217—44no—
139#217—82no—
140#217—82no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146C4×Dic3D812yescomplemented
147#208—41no—
148#208—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×#41—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4#6712yes—
155C23×C4—62no—
156C2×D8—24no—
157C2×M16—61no—
158C4⋊Q8C2×Dic311yescharacteristic, complemented
159#127—21no—
160#127—21no—
161#127C2×Dic312yes—
162#127C2×Dic311yescharacteristic, complemented
163#127#6712yes—
164#124—21no—
165#121—21no—
166#121—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—122no—
170C4×D4—122no—
171C2×C4⋊C4—61no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C42—31no—
176C2×C42—31no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#104—122no—
183#104—122no—
184#102—24no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C4×C8—61no—
188C4×C8C2×Dic312yescomplemented
189#95—32no—
190#95—61no—
191#95—61no—
192#95—61no—
193C23×C3—24no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3#12011yescharacteristic
202C3×Q8—41no—
203C3×Q8—41no—
204C3×D4—44no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—41no—
208C3×D4—42no—
209C3×D4—41no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—82no—
213C3×D4—82no—
214C3×D4—82no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—88no—
243C3×C8—42no—
244C3⋊C8—81no—
245C3⋊C8—81no—
246C24—22no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4#21714yes—
268C22×C4—41no—
269C22×C4—61no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—124no—
273C22×C4—242no—
274C22×C4—242no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278D8—84no—
279M16—121no—
280M16—121no—
281C2×C8—22no—
282C2×C8—121no—
283C2×C8—121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—124no—
291C22⋊C4—242no—
292C22⋊C4—242no—
293C42—32no—
294C42—61no—
295C42—61no—
296C42—61no—
297C42#23711yescharacteristic
298C42#23711yescharacteristic
299C42C22×Dic311yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#38611yescharacteristic
314C22×C3#44711yescharacteristic
315C22×C3#44911yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—81no—
321C3×C4C4×D811yescharacteristic
322C3×C4#44211yescharacteristic
323C3×C4#45911yescharacteristic
324C3×C4#46311yescharacteristic
325Dic3—42no—
326Dic3—81no—
327Dic3—82no—
328C23—24no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23#93711yescharacteristic
337Q8—41no—
338Q8—41no—
339D4—44no—
340D4—42no—
341D4—42no—
342D4—41no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—42no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—81no—
351D4—81no—
352D4—81no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—248no—
368C2×C4D3×D811yescharacteristic
369C2×C4#91111yescharacteristic
370C2×C4#91111yescharacteristic
371C2×C4#91211yescharacteristic
372C2×C4D4×Dic311yescharacteristic
373C2×C4D4×Dic311yescharacteristic
374C2×C4#93511yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4#93911yescharacteristic
377C2×C4C2×#21711yescharacteristic
378C8—42no—
379C8—241no—
380C8—241no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—84no—
384C2×C3—161no—
385C2×C3#187911yescharacteristic
386C2×C3#190411yescharacteristic
387C2×C3#191111yescharacteristic
388C22—21no—
389C22—21no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#515611yescharacteristic
401C22#515811yescharacteristic
402C22#522011yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—81no—
408C4—122no—
409C4—242no—
410C4—241no—
411C4Dic3×D811yescharacteristic
412C4#515011yescharacteristic
413C4#515211yescharacteristic
414C4#515311yescharacteristic
415C3#929911yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—84no—
419C2—161no—
420C2#7185311yescharacteristic
421C2#7203711yescharacteristic
422C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3205C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3205D3
hypocenterC3#9299
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
2. #83172C2semidirect
3. #71816C2semidirect
4. #71814C2semidirect
5. #71759C2semidirect
6. #67738C2semidirect
7. #67737C2semidirect
12. C3×#493C4semidirect
13. C3×#486C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
45. C3×#127C2×C4semidirect
65. C3×C4×C8C2×C4semidirect
80. #493Dic3semidirect
81. #486Dic3semidirect
146. C4×Dic3D8semidirect
158. C4⋊Q8C2×Dic3semidirect
162. #127C2×Dic3semidirect
188. C4×C8C2×Dic3semidirect
415. C3#9299semidirect

Maximal subgroups

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

subgroupindex
C3×#32052
#831722
#718162
#718142
#717592
#677382
#677372
#92993