Finite Group Databasebeta

← #10690834

Subgroups — #10690834

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83269C211yesmaximal, characteristic
2C2×#4833C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4C2×#4547C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic
6#67702C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9130—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×C42×D3C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5229—21no—
15C2×(C2×D3)⋊C8—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×C8×Dic3—21no—
18#4987—21no—
19#4987C412yes—
20#4972—21no—
21#4972C412yes—
22#4935C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4833—22no—
26#4833—22no—
27C3×#332—22no—
28#4550—21no—
29#4547—22no—
30C2×#431—31no—
31#2857—31no—
32#1797—31no—
33C2×#332—31no—
34C2×#332D311yescharacteristic, complemented
35#1296—31no—
36#1296—31no—
37C22×C3×Q8—21no—
38C22×C4×D3C2×C411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50#946—42no—
51#946—42no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3C2×C412yes—
56C2×Dic3⋊C4—22no—
57C2×C4×Dic3C2×C411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C42×D3—21no—
63C42×D3—21no—
64C42×D3—21no—
65C42×D3—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72(C2×D3)⋊C8—24no—
73(C2×D3)⋊C8—42no—
74C8×Dic3—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C22⋊Q8—61no—
81C22×C42—31no—
82#431—62no—
83#431—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×C22⋊C8—61no—
89C2×C22⋊C8—61no—
90C2×C4×C8—61no—
91#391—61no—
92#389—32no—
93#389—61no—
94#384—31no—
95#380—32no—
96#380—61no—
97#336—61no—
98#332—22no—
99#332—62no—
100C23×D3C22⋊C411yescharacteristic
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C22×C3×C4—22no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114C22×Dic3C22⋊C411yescharacteristic
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—22no—
128C2×C4×D3—41no—
129C2×C4×D3D812yes—
130C2×C4×D3SD1612yes—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—44no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—42no—
151Dic3⋊C4—44no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C22×Q8—21no—
165C23×C4—31no—
166C23×C4—31no—
167C23×C4—31no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—61no—
171C22×C8—61no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C22⋊Q8—122no—
177C22⋊Q8—122no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4C4×D312yes—
184C2×C22⋊C4—62no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—61no—
189C2×C42—61no—
190C2×C42—61no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201C22⋊C8—64no—
202C22⋊C8—122no—
203C4×C8—64no—
204#95—62no—
205#95—62no—
206C23×C3C2×C22⋊C411yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3#10212yes—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×Q8—81no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3#10112yes—
248C2×Dic3#10212yes—
249C4×D3—22no—
250C4×D3—22no—
251C4×D3—22no—
252C4×D3—22no—
253C4×D3—42no—
254C4×D3—42no—
255C4×D3—41no—
256C4×D3—41no—
257C4×D3—41no—
258C4×D3—41no—
259C3×C8—42no—
260C3×C8—81no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263C24—31no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C22×C4—22no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—31no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—62no—
290C22×C4—121no—
291C22×C4C2×C4×D311yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—62no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8—121no—
305C2×C8—121no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—122no—
311C4⋊C4—124no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C22⋊C4—124no—
317C42—61no—
318C42—61no—
319C42—61no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42#23511yescharacteristic
324C42D3×D411yescharacteristic
325C42#23711yescharacteristic
326C42C2×#6711yescharacteristic
327C22×C3#38611yescharacteristic
328C22×C3C2×#10111yescharacteristic
329C22×C3#41311yescharacteristic
330C22×C3C2×#10211yescharacteristic
331C22×C3#41611yescharacteristic
332C22×C3#41611yescharacteristic
333C22×C3#41911yescharacteristic
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4#45211yescharacteristic
345C3×C4#45211yescharacteristic
346C3×C4#46511yescharacteristic
347C3×C4#46511yescharacteristic
348C3×C4#46611yescharacteristic
349C3×C4#46611yescharacteristic
350C3×C4#48511yescharacteristic
351C3×C4#48511yescharacteristic
352Dic3—22no—
353Dic3—22no—
354Dic3—41no—
355Dic3—82no—
356C23—32no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23D3×C22⋊C411yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4#67⋊C411yescharacteristic
396C2×C4#88311yescharacteristic
397C2×C4#88411yescharacteristic
398C2×C4D3×D811yescharacteristic
399C2×C4#91211yescharacteristic
400C2×C4D3×SD1611yescharacteristic
401C2×C4#91511yescharacteristic
402C2×C4#91611yescharacteristic
403C2×C4#91611yescharacteristic
404C2×C4#91811yescharacteristic
405C2×C4#91911yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93711yescharacteristic
408C2×C4#93811yescharacteristic
409C8—42no—
410C8—81no—
411C8—122no—
412C8—241no—
413C2×C3#154011yescharacteristic
414C2×C3#154411yescharacteristic
415C2×C3#154611yescharacteristic
416C2×C3#155411yescharacteristic
417C2×C3#187311yescharacteristic
418C2×C3#189311yescharacteristic
419C2×C3#190111yescharacteristic
420D3—42no—
421C22—62no—
422C22—62no—
423C22—62no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—121no—
428C22D3×#10111yescharacteristic
429C22#475111yescharacteristic
430C22D3×#10211yescharacteristic
431C22#477211yescharacteristic
432C22#480211yescharacteristic
433C22#480411yescharacteristic
434C22#495611yescharacteristic
435C4—21no—
436C4—21no—
437C4—62no—
438C4—62no—
439C4—82no—
440C4—121no—
441C4—242no—
442C4#477511yescharacteristic
443C4#477811yescharacteristic
444C4#481411yescharacteristic
445C4#481511yescharacteristic
446C4#515711yescharacteristic
447C4#517111yescharacteristic
448C4#517311yescharacteristic
449C4#518711yescharacteristic
450C3#913011yescharacteristic, complemented, Sylow
451C2—122no—
452C2#6841711yescharacteristic
453C2#6841911yescharacteristic
454C2#6842011yescharacteristic
455C2#6842711yescharacteristic
456C2#7011511yescharacteristic
457C2#7020211yescharacteristic
458C2#7026911yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9130
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. C2×C3×#332C2semidirect
4. C2×#4547C2semidirect
6. #67702C2semidirect
7. #67702C2semidirect
34. C2×#332D3semidirect
450. C3#9130semidirect

Maximal subgroups

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

subgroupindex
#832692
C2×#48332
C2×C3×#3322
C2×#45472
#704752
#677022
#91303