Finite Group Databasebeta

← #10690218

Subgroups — #10690218

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4985C211yesmaximal, characteristic
2#73867C211yesmaximal, characteristic
3C3×#1824C211yesmaximal, characteristic
4#71554C211yesmaximal, characteristic
5C2×#4469C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#9022—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5212C2211yescharacteristic
14C2×C3⋊#95—21no—
15C2×C3⋊#95—21no—
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4985—22no—
19#4985C414yescomplemented
20#4978—22no—
21#4943—21no—
22Dic3×C22⋊C4—21no—
23#4655—22no—
24C3×#340—22no—
25#4529—22no—
26#4469—22no—
27C2×#400—31no—
28#2313—31no—
29#1824—31no—
30#1824D311yescharacteristic
31C2×#324—31no—
32#1294—31no—
33#1294—31no—
34C23×C3×C4—21no—
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×C4⋊Dic3D412yes—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4C2×C412yes—
55C2×Dic3⋊C4D412yes—
56C2×C4×Dic3—22no—
57C22×C3⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3⋊#95—22no—
62C3⋊#95—24no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—24no—
66C3⋊#95—24no—
67C3⋊#95C2×C412yes—
68#830—21no—
69#830—21no—
70C22×C4⋊C4—31no—
71C22×C4⋊C4C2×D311yescharacteristic
72C2×C22⋊C8—31no—
73C2×C22⋊C8C2×D311yescharacteristic
74#400—34no—
75#400—62no—
76#387—62no—
77#387—62no—
78#381—61no—
79#380—31no—
80#380C2×D311yescharacteristic
81C4×C22⋊C4—61no—
82C2×#95—31no—
83C2×#95—61no—
84C2×#95—61no—
85#340—22no—
86#340—62no—
87#324—62no—
88C24×C3C2×D411yescharacteristic
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—42no—
92C22×C3×C4—44no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—42no—
112C22×Dic3—42no—
113C22×Dic3—42no—
114C22×Dic3—41no—
115C22×Dic3—41no—
116C22×Dic3C22⋊C412yes—
117C22×Dic3C22⋊C412yes—
118C2×C3×C8—42no—
119C2×C3×C8—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—42no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127#217—41no—
128#217—41no—
129C4⋊Dic3—42no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134C4×Dic3—82no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—41no—
137C23×C4—21no—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4—61no—
141C23×C4C22×D311yescharacteristic
142C22×C8—21no—
143C22×C8—61no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—32no—
150C2×C4⋊C4—32no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4C4×D312yes—
157C2×C4⋊C4#6712yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—61no—
160C2×C42—62no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165#95—21no—
166#95—32no—
167#95—64no—
168#95—64no—
169#95—62no—
170#95—64no—
171#95—61no—
172#95—61no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3C2×C22⋊C411yescharacteristic
176C23×C3#12011yescharacteristic
177C23×C3#12311yescharacteristic
178C23×C3#13511yescharacteristic
179C23×C3#13511yescharacteristic
180C23×C3#13711yescharacteristic
181C23×C3#13711yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—44no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—82no—
194C2×C3×C4—82no—
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—81no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218C3×C8—81no—
219C3⋊C8—81no—
220C24D3×D411yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—122no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4#23511yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4#23711yescharacteristic
249C22×C4C2×#6711yescharacteristic
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—122no—
253C2×C8—121no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—122no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C42—242no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#38611yescharacteristic
278C22×C3#41611yescharacteristic
279C22×C3#41911yescharacteristic
280C22×C3#45211yescharacteristic
281C22×C3#45211yescharacteristic
282C22×C3#45412yes—
283C22×C3#45812yes—
284C22×C3#48511yescharacteristic
285C22×C3#48511yescharacteristic
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—82no—
291C3×C4—82no—
292Dic3—42no—
293Dic3—42no—
294Dic3—82no—
295Dic3—82no—
296C23—22no—
297C23—22no—
298C23D3×C22⋊C411yescharacteristic
299C23#88311yescharacteristic
300C23#91511yescharacteristic
301C23#91611yescharacteristic
302C23#91811yescharacteristic
303C23#91911yescharacteristic
304C23#93711yescharacteristic
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—241no—
335C2×C4—242no—
336C2×C4—242no—
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C2×C4#93811yescharacteristic
341C8—81no—
342C8—241no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#161512yes—
348C2×C3#161512yes—
349C2×C3#187311yescharacteristic
350C2×C3#188811yescharacteristic
351C2×C3#189011yescharacteristic
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22#480211yescharacteristic
361C22#480411yescharacteristic
362C22#481411yescharacteristic
363C22#481511yescharacteristic
364C22#495611yescharacteristic
365C22#505312yes—
366C22#517311yescharacteristic
367C22#518711yescharacteristic
368C22#520112yes—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4—122no—
376C4—122no—
377C4—242no—
378C4—242no—
379C3#902211yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#6953811yescharacteristic
385C2#7026911yescharacteristic
386C2#7032511yescharacteristic
387C2#7117012yes—
388C2#7117812yes—
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1824C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1824D3
hypocenterC3#9022
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
19. #4985C4semidirect
379. C3#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#49852
#738672
C3×#18242
#715542
C2×#44692
#676752
#676242
#90223