Finite Group Databasebeta

← #10695563

Subgroups — #10695563

Order 768.

768 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2539C211yesmaximal, characteristic
2#82616C211yesmaximal, characteristic, complemented
3#82113C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic, complemented
6#69442C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9940—31nomaximal, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C2×C3×#120—21no—
12C22×C3×C22⋊C4C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C3×#399—21no—
15C3×#392—21no—
16C3×#390—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5247—21no—
19#5215—21no—
20#5211C2211yescharacteristic
21#5210—21no—
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5035—22no—
25#5034—21no—
26#5033—21no—
27#4964—21no—
28#4953—21no—
29#4949—21no—
30#4944—22no—
31#4537—22no—
32#4525—22no—
33#4523—22no—
34#3094—31no—
35#2539D311yescharacteristic
36#2482—31no—
37#1826—31no—
38#1510—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4—22no—
44C22×C3×D4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C3×#123—44no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#120—44no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42D412yes—
66C2×#217—22no—
67C2×#217—21no—
68C2×#217—21no—
69C2×#217—22no—
70#926—21no—
71#926—21no—
72#926—21no—
73#926—21no—
74C2×C4⋊Dic3D411yescharacteristic, complemented
75C2×C4⋊Dic3D411yescharacteristic, complemented
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—21no—
78C2×#208—21no—
79C2×#208—21no—
80C3×#95—21no—
81#832—22no—
82#832—22no—
83C3⋊#95—22no—
84C3⋊#95—21no—
85C3⋊#95D412yescomplemented
86#830—21no—
87#830—21no—
88#830—21no—
89#830D412yescomplemented
90#808—22no—
91#807—22no—
92C2×#123—21no—
93C2×#121C2×D311yescharacteristic
94C2×#120—21no—
95C2×#117—31no—
96C22×C22⋊C4C2×D311yescharacteristic
97#482—62no—
98#481—61no—
99#480—61no—
100#428—61no—
101#418—61no—
102#407—31no—
103#407—31no—
104#399—21no—
105#399—61no—
106#397—61no—
107#392—21no—
108#390—22no—
109#388—62no—
110#385—61no—
111#381—31no—
112#379—61no—
113C4×C22⋊C4C2×D311yescharacteristic
114#339—62no—
115#327—62no—
116#323—62no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—42no—
122C24×C3—42no—
123C24×C3—42no—
124C24×C3C2×D411yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—44no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—44no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—82no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—44no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C42—42no—
164#217—42no—
165#217—44no—
166#217—41no—
167#217—41no—
168#217—82no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173Dic3⋊C4—82no—
174C4×Dic3—41no—
175C4×Dic3—41no—
176#208—42no—
177#208—42no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—41no—
180C25#6711yescharacteristic
181C22×D4—21no—
182C22×D4—22no—
183C22×D4#6711yescharacteristic
184C23×C4#6711yescharacteristic
185C23×C4C22×D311yescharacteristic
186C2×M16—61no—
187C2×M16—61no—
188#123—44no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193#120—44no—
194#117—61no—
195#117—61no—
196#117—61no—
197#117—61no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C42—61no—
218C2×C42#6712yes—
219#107—62no—
220#106—62no—
221#102—62no—
222#102—62no—
223C22⋊C8—32no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227#95—21no—
228#95—32no—
229#95—62no—
230#95—61no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—81no—
251C23×C3—81no—
252C23×C3—88no—
253C23×C3#12011yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3#12111yescharacteristic
256C3×D4—42no—
257C3×D4—81no—
258C3×D4—88no—
259C3×D4—82no—
260C3×D4—82no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—81no—
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12711yescharacteristic
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C3⋊C8—42no—
289C3⋊C8—81no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24C2×#6711yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—82no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4D3×D411yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4#23711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6711yescharacteristic
324M16—122no—
325M16—122no—
326C2×C8—62no—
327C2×C8—121no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—44no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—61no—
335C4⋊C4—242no—
336C22⋊C4—24no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—122no—
346C22⋊C4—124no—
347C22⋊C4—121no—
348C22⋊C4—121no—
349C22⋊C4—242no—
350C42—42no—
351C42—121no—
352C42—121no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—44no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—81no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3#39211yescharacteristic
376C22×C3#45312yes—
377C22×C3#45812yes—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—44no—
383C3×C4—81no—
384C3×C4—81no—
385Dic3—82no—
386Dic3—81no—
387Dic3—81no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—42no—
407C23—81no—
408C23—81no—
409C23—88no—
410C23#93711yescharacteristic
411C23#93911yescharacteristic
412C23#95311yescharacteristic
413D4—42no—
414D4—81no—
415D4—88no—
416D4—82no—
417D4—82no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4#93711yescharacteristic
442C2×C4#93811yescharacteristic
443C2×C4#93911yescharacteristic
444C2×C4#94011yescharacteristic
445C8—122no—
446C8—241no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—82no—
451C2×C3—161no—
452C2×C3#200211yescharacteristic
453C2×C3#201011yescharacteristic
454C2×C3#202211yescharacteristic
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#507712yes—
478C22#522311yescharacteristic
479C22#524312yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—44no—
485C4—81no—
486C4—81no—
487C4—242no—
488C4—241no—
489C4—241no—
490C3#994011yescharacteristic, complemented, Sylow
491C2—22no—
492C2—22no—
493C2—41no—
494C2—82no—
495C2—161no—
496C2#7171511yescharacteristic
497C2#7176911yescharacteristic
498C2#7184911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2539C2
socleC22×C3#392
hypercenter#2539D3
hypocenterC3#9940
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
2. #82616C2semidirect
3. #82113C2semidirect
4. #71674C2semidirect
5. #71539C2semidirect
6. #69442C2semidirect
7. #67616C2semidirect
74. C2×C4⋊Dic3D4semidirect
75. C2×C4⋊Dic3D4semidirect
85. C3⋊#95D4semidirect
89. #830D4semidirect
490. C3#9940semidirect

Maximal subgroups

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

subgroupindex
C3×#25392
#826162
#821132
#716742
#715392
#694422
#676162
#99403