Finite Group Databasebeta

← #10694030

Subgroups — #10694030

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C2×#4525C214yesmaximal, complemented
3#71530C211yesmaximal, characteristic, complemented
4#71529C211yesmaximal, characteristic, complemented
5#9653—31nomaximal, Sylow
6C23×C3×D4C411yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic
8C22×C3×C22⋊C4C412yes—
9C22×C3×C22⋊C4C2212yes—
10C22×C3×C42C2211yescharacteristic
11C3×#390—21no—
12C3×#390—21no—
13C3×#390—21no—
14C3×#390—21no—
15C2×#830C2212yes—
16C2×#830C2212yes—
17C2×C4⋊(C3⋊C8)—21no—
18C2×#803—21no—
19#4525—28no—
20#4524—21no—
21#4524—21no—
22C2×#390D311yescharacteristic
23#1797—31no—
24#1749—31no—
25C2×#323—34no—
26C25×C3C2×C412yes—
27C22×C3×D4—24no—
28C22×C3×D4—21no—
29C22×C3×D4—21no—
30C22×C3×D4—42no—
31C22×C3×D4—42no—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4C2×C412yes—
35C23×C3×C4C2311yescharacteristic
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—42no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—24no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4—44no—
46C2×C3×C42—42no—
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C22×C3⋊C8—22no—
52C22×C3⋊C8—22no—
53#830—22no—
54#830—24no—
55#830D414yescomplemented
56C4⋊(C3⋊C8)—42no—
57#803—42no—
58C23×D4Dic311yescharacteristic
59C22×C4⋊C4Dic311yescharacteristic
60C22×C22⋊C4Dic312yes—
61C22×C22⋊C4C2×D312yes—
62C22×C42C2×D311yescharacteristic
63C2×C4⋊C8—61no—
64C2×C22⋊C8—32no—
65C2×C22⋊C8—32no—
66C2×#97—61no—
67#390—21no—
68#390—21no—
69#390—21no—
70#390—21no—
71#336—61no—
72#336—61no—
73#323—68no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—24no—
77C24×C3—24no—
78C24×C3—42no—
79C24×C3—42no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3M1614yes—
83C24×C3C22×C411yescharacteristic
84C2×C3×D4—24no—
85C2×C3×D4—24no—
86C2×C3×D4—416no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—24no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4112yes—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—44no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C42—22no—
119C3×C42—22no—
120C3×C42—41no—
121C3×C42—41no—
122C2×C3⋊C8—24no—
123C2×C3⋊C8—42no—
124C2×C3⋊C8—42no—
125C2×C3⋊C8—42no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—42no—
128C25C2×Dic312yes—
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—42no—
133C22×D4—42no—
134C23×C4C2×Dic311yescharacteristic
135C23×C4C2×Dic311yescharacteristic
136C23×C4C2×Dic312yes—
137C23×C4C22×D311yescharacteristic
138C22×C8—62no—
139C22×C8—62no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—42no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—44no—
150C2×C42—42no—
151C2×C42#6711yescharacteristic
152C2×C42#6711yescharacteristic
153C2×C42#6711yescharacteristic
154C2×C42#6711yescharacteristic
155C4⋊C8—122no—
156C22⋊C8—34no—
157C22⋊C8—62no—
158C22⋊C8—64no—
159#97—122no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—24no—
165C23×C3—28no—
166C23×C3—44no—
167C23×C3—44no—
168C23×C3—44no—
169C23×C3—44no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—44no—
177C23×C3—84no—
178C23×C3—84no—
179C23×C3—84no—
180C23×C3—84no—
181C23×C3C2×C22⋊C411yescharacteristic
182C23×C3C4×D412yes—
183C23×C3C2×M1612yes—
184C23×C3#13112yes—
185C3×D4—48no—
186C3×D4—48no—
187C2×C3×C4—28no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—44no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—84no—
208C2×C3×C4#9912yes—
209C2×C3×C4#10012yes—
210C2×C3×C4#10012yes—
211C2×C3×C4#12112yes—
212C2×C3×C4#12411yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C3⋊C8—44no—
215C3⋊C8—82no—
216C24—22no—
217C24—22no—
218C24—24no—
219C24—24no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24#20814yes—
225C24C22×Dic311yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—416no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4#21711yescharacteristic
246C22×C4#21711yescharacteristic
247C22×C4#21711yescharacteristic
248C22×C4#21711yescharacteristic
249C22×C4D3×D412yes—
250C22×C4#23712yes—
251C22×C4C2×#6711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—64no—
254C2×C8—122no—
255C2×C8—122no—
256C2×C8—122no—
257C2×C8—122no—
258C2×C8—122no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—88no—
280C22×C3—88no—
281C22×C3—82no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3#39011yescharacteristic
286C22×C3#40711yescharacteristic
287C22×C3#40811yescharacteristic
288C22×C3C2×#9911yescharacteristic
289C22×C3#41011yescharacteristic
290C22×C3C2×#10011yescharacteristic
291C22×C3C2×#10011yescharacteristic
292C22×C3#43514yes—
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—41no—
300C3×C4—82no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—24no—
306C23—28no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—84no—
319C23—84no—
320C23—84no—
321C23—84no—
322C23C2×#20812yes—
323C23D4×Dic312yes—
324C23#94812yes—
325C23C2×#21711yescharacteristic
326D4—48no—
327D4—48no—
328C2×C4—28no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—84no—
349C2×C4#83312yes—
350C2×C4#83312yes—
351C2×C4#83412yes—
352C2×C4#93611yescharacteristic
353C2×C4#93812yes—
354C2×C4#94011yescharacteristic
355C8—124no—
356C8—242no—
357C2×C3—24no—
358C2×C3—84no—
359C2×C3#146211yescharacteristic
360C2×C3#146611yescharacteristic
361C2×C3#146911yescharacteristic
362C2×C3#147111yescharacteristic
363C2×C3#195511yescharacteristic
364C2×C3#196411yescharacteristic
365C2×C3#196511yescharacteristic
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—88no—
376C22—88no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22#501114yes—
382C22#520711yescharacteristic
383C22C2×#83311yescharacteristic
384C22C2×#83311yescharacteristic
385C22C2×#83411yescharacteristic
386C22#522111yescharacteristic
387C22#523411yescharacteristic
388C22#524011yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—82no—
397C3#965311yescharacteristic, complemented, Sylow
398C2—24no—
399C2—84no—
400C2#7153911yescharacteristic
401C2#7160311yescharacteristic
402C2#7161511yescharacteristic
403C2#7167611yescharacteristic
404C2#7168611yescharacteristic
405C2#7172011yescharacteristic
406C2#7176611yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#390C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C4
cosocleC23×C3×C4C23
hypercenterC2×#390D3
hypocenterC3#9653
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#4525C2semidirect
3. #71530C2semidirect
4. #71529C2semidirect
55. #830D4semidirect
397. C3#9653semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×#45252
#715302
#715292
#96533