Finite Group Databasebeta

← #10690435

Subgroups — #10690435

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4694C211yesmaximal, characteristic
2#73653C211yesmaximal, characteristic
3#70456C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic
5#69490C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9057—31nomaximal, Sylow
9C2×#893—21no—
10C2×#874C2211yescharacteristic
11C2×C4×D12—21no—
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×#821—22no—
17C2×#816—21no—
18C2×#816—21no—
19#4988—21no—
20#4939—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23#4694—22no—
24#4694C414yescomplemented
25#4665—22no—
26#4529—21no—
27#4529—21no—
28#4485—22no—
29C2×#486—31no—
30#2366—31no—
31#1988—31no—
32#1981—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—
56#893—42no—
57#893—42no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61#874D412yes—
62C4×D12—42no—
63C4×D12—42no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869C2×C412yescomplemented
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4×C8—21no—
71C3×C4×C8C2×C412yescomplemented
72C3×#95—21no—
73C3⋊#95—21no—
74#821—28no—
75#816—42no—
76#816—42no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C2×C4⋊Q8—31no—
80C2×#127—31no—
81C2×C4×D4—61no—
82C2×C4×D4—61no—
83#486—34no—
84#486—62no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×#103—61no—
88C2×#103—61no—
89C2×#102—62no—
90C2×C4×C8C2×D311yescharacteristic
91#390—61no—
92#390—61no—
93#386—62no—
94C4×C4⋊C4—31no—
95C4×C4⋊C4C2×D311yescharacteristic
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—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×Dic6—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4SD1612yes—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42C22×C411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148C3×C42#4111yescharacteristic
149(C2×D3)⋊C4—44no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—82no—
152C4⋊Dic3—21no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4⋊Dic3D812yes—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×Q8—61no—
166C22×D4—32no—
167C22×D4—61no—
168C22×D4—61no—
169C23×C4—62no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—61no—
173C4⋊Q8—32no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177#127—32no—
178#127—32no—
179#127—61no—
180#127—61no—
181C4×D4—122no—
182C4×D4—122no—
183C4×D4—122no—
184C4×D4—122no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4D1211yescharacteristic
190C2×C4⋊C4D1211yescharacteristic
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42—21no—
195C2×C42—61no—
196C2×C42C22×D311yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#103—122no—
202#103—122no—
203#102—68no—
204C4×C8—21no—
205C4×C8C4×D312yescomplemented
206#95—21no—
207#95—61no—
208C23×C3#12011yescharacteristic
209C22×D3—24no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×Dic3—22no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247D12—44no—
248D12—42no—
249D12—42no—
250D12—42no—
251D12—42no—
252D12—42no—
253D12—42no—
254D12—42no—
255D12—42no—
256D12—82no—
257C4×D3—88no—
258Dic6—42no—
259Dic6—42no—
260Dic6—81no—
261Dic6—81no—
262C3×C8—42no—
263C3×C8—81no—
264C3⋊C8—81no—
265C24—62no—
266C2×Q8—62no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×Q8—121no—
271C2×Q8—121no—
272C2×D4—34no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—122no—
279C2×D4—122no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—124no—
293C22×C4—124no—
294C22×C4—121no—
295C22×C4—242no—
296C22×C4C2×D1211yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—121no—
306C2×C8—121no—
307C2×C8—121no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—32no—
312C4⋊C4—61no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4#20412yes—
320C22⋊C4—124no—
321C22⋊C4—124no—
322C22⋊C4—242no—
323C42—41no—
324C42—41no—
325C42—121no—
326C42—121no—
327C42C2×C4×D311yescharacteristic
328C42#23511yescharacteristic
329C42D3×D411yescharacteristic
330C42#23911yescharacteristic
331C22×C3#38611yescharacteristic
332C22×C3#44711yescharacteristic
333C22×C3#44911yescharacteristic
334C22×C3#45011yescharacteristic
335C22×C3#45211yescharacteristic
336C22×C3#45511yescharacteristic
337C22×C3#45511yescharacteristic
338C2×D3—44no—
339C2×D3—44no—
340C2×D3—44no—
341C2×D3—82no—
342C2×D3—82no—
343C2×D3—82no—
344C2×D3—82no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—42no—
348C3×C4—81no—
349C3×C4C4×D811yescharacteristic
350C3×C4C4×SD1611yescharacteristic
351C3×C4#44011yescharacteristic
352C3×C4#44211yescharacteristic
353C3×C4#45911yescharacteristic
354C3×C4#46111yescharacteristic
355C3×C4#46311yescharacteristic
356C3×C4#46311yescharacteristic
357Dic3—42no—
358Dic3—82no—
359Dic3—81no—
360C23—64no—
361C23—122no—
362C23—122no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23—122no—
367C23#88211yescharacteristic
368Q8—122no—
369Q8—122no—
370Q8—241no—
371Q8—241no—
372D4—124no—
373D4—122no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—122no—
378D4—122no—
379D4—122no—
380D4—122no—
381D4—242no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—248no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4C4×D1211yescharacteristic
401C2×C4D3×C22⋊C411yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#89311yescharacteristic
404C2×C4#89411yescharacteristic
405C2×C4#89511yescharacteristic
406C2×C4C2×#20411yescharacteristic
407C2×C4#90411yescharacteristic
408C2×C4#90811yescharacteristic
409C2×C4#90911yescharacteristic
410C2×C4D3×D811yescharacteristic
411C2×C4#91411yescharacteristic
412C2×C4#91411yescharacteristic
413C2×C4#91911yescharacteristic
414C8—42no—
415C8—81no—
416C8—241no—
417C2×C3#163911yescharacteristic
418C2×C3#164011yescharacteristic
419C2×C3#164111yescharacteristic
420C2×C3#164211yescharacteristic
421C2×C3#187711yescharacteristic
422C2×C3#187911yescharacteristic
423C2×C3#188511yescharacteristic
424D3—84no—
425C22—124no—
426C22—124no—
427C22—124no—
428C22—242no—
429C22—242no—
430C22—242no—
431C22—242no—
432C22#466511yescharacteristic
433C22#473111yescharacteristic
434C22#473311yescharacteristic
435C22#477311yescharacteristic
436C22#481011yescharacteristic
437C22#482511yescharacteristic
438C22#482511yescharacteristic
439C4—21no—
440C4—21no—
441C4—42no—
442C4—81no—
443C4—122no—
444C4—242no—
445C4—241no—
446C4C4×#20411yescharacteristic
447C4#471011yescharacteristic
448C4#478611yescharacteristic
449C4#481911yescharacteristic
450C4#484111yescharacteristic
451C4#484411yescharacteristic
452C4#486911yescharacteristic
453C4#487211yescharacteristic
454C3#905711yescharacteristic, complemented, Sylow
455C2—244no—
456C2#6818211yescharacteristic
457C2#6819011yescharacteristic
458C2#6824811yescharacteristic
459C2#6825411yescharacteristic
460C2#6867211yescharacteristic
461C2#6875911yescharacteristic
462C2#6889711yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9057
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. #70456C2semidirect
5. #69490C2semidirect
6. C3×#1279C2semidirect
7. #67655C2semidirect
24. #4694C4semidirect
35. #1279D3semidirect
67. #869C2×C4semidirect
71. C3×C4×C8C2×C4semidirect
205. C4×C8C4×D3semidirect
454. C3#9057semidirect

Maximal subgroups

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

subgroupindex
C2×#46942
#736532
#704562
C2×#44852
#694902
C3×#12792
#676552
#90573