Finite Group Databasebeta

← #10690480

Subgroups — #10690480

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4693C211yesmaximal, characteristic
2#73653C211yesmaximal, characteristic
3#70459C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic
5#69489C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9064—31nomaximal, Sylow
9C22×#204—22no—
10C2×#893—21no—
11C2×#874C2211yescharacteristic
12C2×C4×D12—21no—
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17#4988—21no—
18#4939—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4693—22no—
22#4693C414yescomplemented
23#4665—22no—
24#4551—21no—
25#4551—21no—
26#4529—21no—
27#4529—21no—
28#4485—22no—
29C2×#492—31no—
30#2366—31no—
31#1987—31no—
32#1987—31no—
33C2×#331—31no—
34#1279—31no—
35#1279D311yescharacteristic, complemented
36C22×D12—21no—
37C22×D12—21no—
38C22×D12D412yes—
39C22×C4×D3—22no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C2×#204—28no—
57C2×#204—44no—
58C2×#204—44no—
59#893—42no—
60#893—42no—
61#874—21no—
62#874—21no—
63#874C2×C412yes—
64#874D412yes—
65C4×D12—42no—
66C4×D12—42no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869C2×C412yescomplemented
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8C2×C412yescomplemented
75C3×#95—21no—
76C3⋊#95—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C22×SD16—62no—
80C2×C4⋊Q8—31no—
81C2×#127—31no—
82C2×C4×D4—61no—
83C2×C4×D4—61no—
84#492—34no—
85#492—62no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8C2×D311yescharacteristic
88C2×C4×C8C2×D311yescharacteristic
89#390—61no—
90#390—61no—
91#386—62no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4C2×D311yescharacteristic
94#340—61no—
95#340—61no—
96#340—61no—
97#340—61no—
98#331—62no—
99C23×D3—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—41no—
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—42no—
122C2×D12—42no—
123C2×D12C22⋊C414yes—
124C2×C4×D3—44no—
125C2×C4×D3—44no—
126C2×C4×D3—82no—
127C2×Dic6—22no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—41no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4SD1612yescomplemented
141C3×C42—41no—
142C3×C42—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147(C2×D3)⋊C4—44no—
148(C2×D3)⋊C4—44no—
149(C2×D3)⋊C4—82no—
150C4⋊Dic3—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3SD1612yescomplemented
158C4×Dic3—41no—
159C4×Dic3—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163#204—416no—
164C22×Q8—61no—
165C22×D4—32no—
166C22×D4—61no—
167C22×D4—61no—
168C23×C4—62no—
169C2×SD16—68no—
170C2×SD16—124no—
171C2×SD16—124no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—61no—
175C4⋊Q8—32no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179#127—32no—
180#127—32no—
181#127—61no—
182#127—61no—
183C4×D4—122no—
184C4×D4—122no—
185C4×D4—122no—
186C4×D4—122no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4D1211yescharacteristic
192C2×C4⋊C4D1211yescharacteristic
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C42—21no—
197C2×C42—61no—
198C2×C42C22×D311yescharacteristic
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4×C8—21no—
204C4×C8C4×D312yescomplemented
205#95—21no—
206#95—61no—
207C23×C3#12011yescharacteristic
208C22×D3—24no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
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×SD1611yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246D12—44no—
247D12—42no—
248D12—42no—
249D12—42no—
250D12—42no—
251D12—42no—
252D12—42no—
253D12—42no—
254D12—42no—
255D12—82no—
256C4×D3—88no—
257Dic6—44no—
258Dic6—82no—
259C3×C8—24no—
260C3×C8—81no—
261C3⋊C8—81no—
262C24—62no—
263C2×Q8—62no—
264C2×Q8—122no—
265C2×Q8—122no—
266C2×Q8—121no—
267C2×D4—34no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—122no—
274C2×D4—122no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—124no—
288C22×C4—124no—
289C22×C4—121no—
290C22×C4—242no—
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294SD16—1216no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—32no—
308C4⋊C4—61no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4#20412yescomplemented
316C22⋊C4—124no—
317C22⋊C4—124no—
318C22⋊C4—242no—
319C42—41no—
320C42—41no—
321C42—121no—
322C42—121no—
323C42C2×C4×D311yescharacteristic
324C42#23511yescharacteristic
325C42D3×D411yescharacteristic
326C42#23911yescharacteristic
327C22×C3#38611yescharacteristic
328C22×C3#45011yescharacteristic
329C22×C3#45011yescharacteristic
330C22×C3#45211yescharacteristic
331C22×C3#45211yescharacteristic
332C22×C3#45311yescharacteristic
333C22×C3#45311yescharacteristic
334C2×D3—44no—
335C2×D3—44no—
336C2×D3—44no—
337C2×D3—82no—
338C2×D3—82no—
339C2×D3—82no—
340C2×D3—82no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—42no—
344C3×C4—81no—
345C3×C4C4×SD1611yescharacteristic
346C3×C4C4×SD1611yescharacteristic
347C3×C4#44011yescharacteristic
348C3×C4#44011yescharacteristic
349C3×C4#46111yescharacteristic
350C3×C4#46111yescharacteristic
351C3×C4#46311yescharacteristic
352C3×C4#46311yescharacteristic
353Dic3—42no—
354Dic3—81no—
355Dic3—82no—
356C23—64no—
357C23—122no—
358C23—122no—
359C23—122no—
360C23—122no—
361C23—122no—
362C23—122no—
363C23#88211yescharacteristic
364Q8—124no—
365Q8—242no—
366D4—124no—
367D4—122no—
368D4—122no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—122no—
374D4—122no—
375D4—242no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—62no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—248no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4C4×D1211yescharacteristic
395C2×C4D3×C22⋊C411yescharacteristic
396C2×C4#88411yescharacteristic
397C2×C4#89311yescharacteristic
398C2×C4#89411yescharacteristic
399C2×C4#89511yescharacteristic
400C2×C4C2×#20411yescharacteristic
401C2×C4#90411yescharacteristic
402C2×C4#90811yescharacteristic
403C2×C4#90911yescharacteristic
404C2×C4#91111yescharacteristic
405C2×C4D3×SD1611yescharacteristic
406C2×C4#91611yescharacteristic
407C2×C4#91811yescharacteristic
408C8—24no—
409C8—81no—
410C8—241no—
411C2×C3#162911yescharacteristic
412C2×C3#163011yescharacteristic
413C2×C3#163111yescharacteristic
414C2×C3#163111yescharacteristic
415C2×C3#187711yescharacteristic
416C2×C3#187711yescharacteristic
417C2×C3#188711yescharacteristic
418D3—84no—
419C22—124no—
420C22—124no—
421C22—124no—
422C22—242no—
423C22—242no—
424C22—242no—
425C22—242no—
426C22#466511yescharacteristic
427C22#473111yescharacteristic
428C22#473311yescharacteristic
429C22#477611yescharacteristic
430C22#480811yescharacteristic
431C22#482211yescharacteristic
432C22#482211yescharacteristic
433C4—21no—
434C4—21no—
435C4—42no—
436C4—81no—
437C4—122no—
438C4—242no—
439C4—241no—
440C4C4×#20411yescharacteristic
441C4#471011yescharacteristic
442C4#478711yescharacteristic
443C4#481811yescharacteristic
444C4#484111yescharacteristic
445C4#484411yescharacteristic
446C4#485211yescharacteristic
447C4#488911yescharacteristic
448C3#906411yescharacteristic, complemented, Sylow
449C2—244no—
450C2#6818811yescharacteristic
451C2#6819511yescharacteristic
452C2#6826011yescharacteristic
453C2#6826111yescharacteristic
454C2#6867211yescharacteristic
455C2#6876011yescharacteristic
456C2#6889611yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9064
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #70459C2semidirect
5. #69489C2semidirect
6. C3×#1279C2semidirect
7. #67655C2semidirect
22. #4693C4semidirect
35. #1279D3semidirect
70. #869C2×C4semidirect
74. C3×C4×C8C2×C4semidirect
140. C3×C4⋊C4SD16semidirect
157. C4⋊Dic3SD16semidirect
204. C4×C8C4×D3semidirect
315. C4⋊C4#204semidirect
448. C3#9064semidirect

Maximal subgroups

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

subgroupindex
C2×#46932
#736532
#704592
C2×#44852
#694892
C3×#12792
#676552
#90643