Finite Group Databasebeta

← #10690230

Subgroups — #10690230

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74004C211yesmaximal, characteristic
2C2×#4677C211yesmaximal, characteristic
3#69925C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic
5#69502C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9024—31nomaximal, Sylow
9C22×(C2×D3)⋊C4—21no—
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C22×C4⋊Dic3—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4990—22no—
19#4955—21no—
20Dic3×C22⋊C4—21no—
21#4677—22no—
22#4677C414yescomplemented
23#4670—22no—
24#4551—22no—
25#4529—22no—
26#4468—22no—
27C2×#399—31no—
28#2312—31no—
29#1824—31no—
30#1824—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×#217—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—24no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—44no—
51C2×(C2×D3)⋊C4C2×C412yes—
52C2×(C2×D3)⋊C4D412yes—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—44no—
60C2×C4⋊Dic3C2×C412yes—
61C2×C4⋊Dic3D412yes—
62C2×C4×Dic3—22no—
63C22×C3⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95C2×C412yes—
69C3⋊#95—22no—
70C3⋊#95—21no—
71#830—21no—
72#830—21no—
73C22×C4⋊C4—31no—
74C22×C4⋊C4—61no—
75C22×C22⋊C4—31no—
76C22×C22⋊C4—61no—
77C2×C22⋊C8—31no—
78C2×C22⋊C8C2×D311yescharacteristic
79#399—34no—
80#399—62no—
81#388—62no—
82#388—62no—
83#380—31no—
84#380C2×D311yescharacteristic
85#379—61no—
86C4×C22⋊C4—61no—
87C2×#95C2×D311yescharacteristic
88#340—62no—
89#340—62no—
90#323—62no—
91C24×C3C2×D411yescharacteristic
92C23×D3—21no—
93C23×D3—21no—
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—42no—
99C23×D3—42no—
100C23×D3C22⋊C412yes—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—22no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3C22⋊C412yes—
124C2×C3×C8—42no—
125C2×C3×C8—41no—
126C3×C4⋊C4—42no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129#217—41no—
130#217—41no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—48no—
136(C2×D3)⋊C4—82no—
137(C2×D3)⋊C4—82no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—48no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—42no—
144C4×Dic3—82no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147C25—31no—
148C23×C4—21no—
149C23×C4—31no—
150C23×C4—61no—
151C23×C4C22×D311yescharacteristic
152C22×C8—21no—
153C22×C8—61no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—64no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—124no—
163C2×C4⋊C4D1212yes—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—32no—
166C2×C22⋊C4—32no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—64no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—124no—
176C2×C42—62no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C22⋊C8—61no—
181#95—21no—
182#95—21no—
183#95—62no—
184#95—61no—
185#95C4×D312yes—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3#12011yescharacteristic
190C23×C3#12311yescharacteristic
191C23×C3#13511yescharacteristic
192C23×C3#13511yescharacteristic
193C23×C3#13711yescharacteristic
194C23×C3#13711yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—42no—
199C22×D3—42no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—82no—
211C22×D3—82no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×Dic3—22no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—81no—
240C2×Dic3—82no—
241C2×Dic3—82no—
242C3×C8—81no—
243C3⋊C8—81no—
244C24—32no—
245C24—61no—
246C24—61no—
247C24—61no—
248C24—61no—
249C24—61no—
250C24—61no—
251C24—122no—
252C24—122no—
253C24D3×D411yescharacteristic
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—32no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—122no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4C2×C4×D311yescharacteristic
272C22×C4C2×D1211yescharacteristic
273C22×C4#23511yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4#23911yescharacteristic
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—122no—
280C2×C8—121no—
281C4⋊C4—42no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—122no—
285C4⋊C4—128no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C22⋊C4—62no—
294C22⋊C4—128no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—242no—
298C22⋊C4—242no—
299C42—242no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3#38611yescharacteristic
309C22×C3#41611yescharacteristic
310C22×C3#41911yescharacteristic
311C22×C3#45211yescharacteristic
312C22×C3#45211yescharacteristic
313C22×C3#45312yes—
314C22×C3#45812yes—
315C22×C3#48511yescharacteristic
316C22×C3#48511yescharacteristic
317C2×D3—44no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—42no—
321C2×D3—82no—
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—82no—
325C2×D3—82no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—82no—
332Dic3—42no—
333Dic3—82no—
334Dic3—82no—
335C23—22no—
336C23—22no—
337C23—62no—
338C23—62no—
339C23—62no—
340C23—122no—
341C23—122no—
342C23—124no—
343C23—124no—
344C23—121no—
345C23—121no—
346C23—122no—
347C23—121no—
348C23—121no—
349C23—122no—
350C23—241no—
351C23—241no—
352C23—242no—
353C23—242no—
354C23D3×C22⋊C411yescharacteristic
355C23#88211yescharacteristic
356C23#89411yescharacteristic
357C23#91211yescharacteristic
358C23#91511yescharacteristic
359C23#91611yescharacteristic
360C23#91811yescharacteristic
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—62no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—241no—
385C2×C4—242no—
386C2×C4—242no—
387C2×C4C4×D1211yescharacteristic
388C2×C4#88411yescharacteristic
389C2×C4#89311yescharacteristic
390C2×C4#89511yescharacteristic
391C8—81no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3#161812yes—
398C2×C3#161812yes—
399C2×C3#187311yescharacteristic
400C2×C3#188711yescharacteristic
401C2×C3#189011yescharacteristic
402D3—82no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—124no—
412C22—122no—
413C22—122no—
414C22—122no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22#466511yescharacteristic
421C22#474212yes—
422C22#477211yescharacteristic
423C22#478311yescharacteristic
424C22#480211yescharacteristic
425C22#480811yescharacteristic
426C22#482212yes—
427C22#486811yescharacteristic
428C22#488811yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—44no—
434C4—82no—
435C4—122no—
436C4—242no—
437C4—242no—
438C3#902411yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—242no—
444C2#6829112yes—
445C2#6832312yes—
446C2#6876011yescharacteristic
447C2#6887911yescharacteristic
448C2#6896311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9024
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69925C2semidirect
5. #69502C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
22. #4677C4semidirect
33. #1294D3semidirect
438. C3#9024semidirect

Maximal subgroups

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

subgroupindex
#740042
C2×#46772
#699252
C2×#44682
#695022
C3×#12942
#676122
#90243