Finite Group Databasebeta

← #10690736

Subgroups — #10690736

Order 768.

560 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78602C211yesmaximal, characteristic
2#73446C211yesmaximal, characteristic
3C2×#4469C211yesmaximal, characteristic, complemented
4C2×#4469C211yesmaximal, characteristic, complemented
5#69919C211yesmaximal, characteristic
6C3×#1282C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9110—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#886C411yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#830C2211yescharacteristic
16C2×#830C2211yescharacteristic
17#5111—22no—
18#4958C411yescharacteristic
19#4950C411yescharacteristic
20#4943C411yescharacteristic
21#4940—21no—
22#4928—21no—
23#4730—22no—
24#4726—21no—
25#4719—21no—
26#4668—21no—
27#4653—21no—
28#4469—21no—
29#4469—22no—
30#4469—22no—
31#4468—21no—
32#2973—31no—
33#2322—31no—
34#1459—31no—
35C2×#324—31no—
36C2×#324—31no—
37#1282—31no—
38#1282D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67C2×C411yescharacteristic
41C23×Dic3C2×C411yescharacteristic
42C23×Dic3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4C2×C411yescharacteristic
48C2×C3×C42—21no—
49C2×#217C2×C411yescharacteristic
50C4×#67—21no—
51C4×#67—21no—
52C4×#67—21no—
53C4×#67—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4D411yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4D411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C2×ΓL(1,𝔽25)—22no—
70#886—22no—
71#886—22no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85(C2×D3)⋊C8—22no—
86#815—22no—
87Dic3⋊C8—22no—
88C2×#123—31no—
89C2×C4×D4—31no—
90C22×C4⋊C4—31no—
91#435—62no—
92#435—62no—
93#432—61no—
94#407—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#389—61no—
100#388—61no—
101#387—61no—
102#386—31no—
103#385—31no—
104#385—61no—
105#381—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#324—62no—
108#324—61no—
109#324—62no—
110#323—61no—
111C24×C3C22×C411yescharacteristic
112C23×D3C22⋊C411yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—81no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—21no—
123C2×#67—21no—
124C2×#67—21no—
125C2×#67—21no—
126C2×#67—21no—
127C2×#67—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—81no—
138C22×Dic3C22⋊C411yescharacteristic
139C22×Dic3C22⋊C411yescharacteristic
140C22×Dic3C22⋊C411yescharacteristic
141C22×Dic3M1612yescomplemented
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—81no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C2×C3×C8—81no—
148C2×C3×C8—81no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4M1612yescomplemented
153C3×C42—81no—
154#217—21no—
155#217—21no—
156(C2×D3)⋊C4—24no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—81no—
160C4⋊Dic3—24no—
161C4⋊Dic3—81no—
162Dic3⋊C4—21no—
163Dic3⋊C4—21no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—81no—
173C2×C3⋊C8—81no—
174ΓL(1,𝔽25)—82no—
175C22×D4—31no—
176C23×C4—31no—
177C23×C4—31no—
178C23×C4—31no—
179C23×C4C22×D311yescharacteristic
180C2×M16—62no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—61no—
184C22×C8—61no—
185#123—62no—
186#123—62no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4C4×D311yescharacteristic
205C2×C22⋊C4C4×D311yescharacteristic
206C2×C42—21no—
207C2×C42—31no—
208C4⋊C8—62no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—32no—
213C22⋊C8—62no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8D1212yes—
218#97—62no—
219#95—21no—
220#95—61no—
221#95—61no—
222#95—61no—
223#95—61no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3C2×C22⋊C411yescharacteristic
227C23×C3C4×D411yescharacteristic
228C23×C3C4×D411yescharacteristic
229C23×C3C2×M1611yescharacteristic
230C23×C3C2×M1611yescharacteristic
231C23×C3#13111yescharacteristic
232C23×C3#13111yescharacteristic
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3—41no—
237C22×D3—41no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254#67—41no—
255#67—41no—
256#67—41no—
257#67—41no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—22no—
263C2×Dic3—21no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C4×D3—42no—
278C4×D3—81no—
279C3×C8—42no—
280C3×C8—81no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—31no—
284C24C2×C4×D311yescharacteristic
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—31no—
294C22×C4—32no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—81no—
309C22×C4—241no—
310C22×C4—241no—
311C22×C4C2×D1211yescharacteristic
312C22×C4#23511yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4#23911yescharacteristic
317M16—242no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C2×C8—81no—
322C2×C8—81no—
323C2×C8—121no—
324C2×C8—241no—
325C2×C8—241no—
326C4⋊C4—64no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C4⋊C4—241no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—64no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—61no—
341C22⋊C4—61no—
342C22⋊C4—241no—
343C22⋊C4ΓL(1,𝔽25)12yescomplemented
344C42—61no—
345C42—61no—
346C42—81no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3#38611yescharacteristic
358C22×C3#40711yescharacteristic
359C22×C3#40811yescharacteristic
360C22×C3#43511yescharacteristic
361C22×C3#43511yescharacteristic
362C22×C3#43611yescharacteristic
363C22×C3#43611yescharacteristic
364C2×D3—42no—
365C2×D3—41no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—81no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—42no—
374C3×C4—81no—
375Dic3—42no—
376Dic3—41no—
377Dic3—41no—
378Dic3—41no—
379Dic3—81no—
380C23—41no—
381C23—41no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—121no—
386C23—121no—
387C23C4×D1211yescharacteristic
388C23D3×C22⋊C411yescharacteristic
389C23#89311yescharacteristic
390C23C2×ΓL(1,𝔽25)11yescharacteristic
391C23#90111yescharacteristic
392C23D3×M1611yescharacteristic
393C23#90711yescharacteristic
394D4—121no—
395D4—121no—
396D4—121no—
397D4—121no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—122no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—241no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4#88211yescharacteristic
432C2×C4#88411yescharacteristic
433C2×C4#89411yescharacteristic
434C2×C4#89511yescharacteristic
435C8—42no—
436C8—81no—
437C8—122no—
438C8—241no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#187011yescharacteristic
446C2×C3#191311yescharacteristic
447C2×C3#191511yescharacteristic
448D3—81no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—122no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—241no—
464C22#466511yescharacteristic
465C22#468811yescharacteristic
466C22#470611yescharacteristic
467C22#472611yescharacteristic
468C22#472711yescharacteristic
469C22#483611yescharacteristic
470C22#483811yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—42no—
476C4—81no—
477C4—122no—
478C4—121no—
479C4—121no—
480C4—121no—
481C4—241no—
482C3#911011yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—241no—
490C2#6866911yescharacteristic
491C2#6896411yescharacteristic
492C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC22×#67C2×C4
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9110
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. C2×#4469C2semidirect
4. C2×#4469C2semidirect
6. C3×#1282C2semidirect
7. #67617C2semidirect
38. #1282D3semidirect
141. C22×Dic3M16semidirect
152. C3×C22⋊C4M16semidirect
343. C22⋊C4ΓL(1,𝔽25)semidirect
482. C3#9110semidirect

Maximal subgroups

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

subgroupindex
#786022
#734462
C2×#44692
#699192
C3×#12822
#676172
#91103