Finite Group Databasebeta

← #10694012

Subgroups — #10694012

Order 768.

936 conjugacy classes of subgroups.

The subgroup lattice has 426 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
4C4×#830C211yesmaximal, characteristic, complemented
5#9649—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×C4×C3⋊C8—21no—
19#4525—28no—
20#4524—21no—
21#4524—21no—
22C2×#390D311yescharacteristic
23#1797—31no—
24C4×C22⋊C8—31no—
25C2×#323—34no—
26C25×C3C2×C412yes—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—42no—
31C22×C3×D4—42no—
32C22×C3×D4C814yescomplemented
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4C2×C412yes—
36C23×C3×C4C2311yescharacteristic
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—42no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4—24no—
46C2×C3×C22⋊C4—44no—
47C2×C3×C42—42no—
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C22×C3⋊C8—22no—
53C22×C3⋊C8—22no—
54#830—22no—
55#830—24no—
56#830D414yescomplemented
57C4⋊(C3⋊C8)—42no—
58C4×C3⋊C8—42no—
59C23×D4Dic311yescharacteristic
60C22×C4⋊C4Dic311yescharacteristic
61C22×C22⋊C4Dic312yes—
62C22×C22⋊C4C2×D312yes—
63C22×C42C2×D311yescharacteristic
64C2×C4⋊C8—61no—
65C2×C22⋊C8—32no—
66C2×C22⋊C8—32no—
67C2×C4×C8—61no—
68#390—21no—
69#390—21no—
70#390—21no—
71#390—21no—
72#336—61no—
73#336—61no—
74#323—68no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—24no—
78C24×C3—24no—
79C24×C3—42no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3—42no—
83C24×C3C2×C814yes—
84C24×C3C22×C411yescharacteristic
85C2×C3×D4—24no—
86C2×C3×D4—24no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—48no—
90C2×C3×D4—48no—
91C2×C3×D4—42no—
92C2×C3×D4—42no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—24no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C2×C812yes—
109C22×C3×C4M1612yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4112yes—
114C3×C4⋊C4—42no—
115C3×C4⋊C4—42no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—44no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—42no—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42—41no—
124C3×C42—41no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—42no—
130C2×C3⋊C8—42no—
131C25C2×Dic312yes—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C3⋊C814yescomplemented
138C23×C4C2×Dic311yescharacteristic
139C23×C4C2×Dic311yescharacteristic
140C23×C4C2×Dic312yes—
141C23×C4C22×D311yescharacteristic
142C22×C8—62no—
143C22×C8—62no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—42no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—44no—
154C2×C42—42no—
155C2×C42#6711yescharacteristic
156C2×C42#6711yescharacteristic
157C2×C42#6711yescharacteristic
158C2×C42#6711yescharacteristic
159C4⋊C8—122no—
160C22⋊C8—34no—
161C22⋊C8—62no—
162C22⋊C8—64no—
163C4×C8—122no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—24no—
169C23×C3—28no—
170C23×C3—44no—
171C23×C3—44no—
172C23×C3—44no—
173C23×C3—44no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—44no—
181C23×C3—84no—
182C23×C3—84no—
183C23×C3—84no—
184C23×C3—84no—
185C23×C3C2×C22⋊C411yescharacteristic
186C23×C3C4×D412yes—
187C23×C3C22×C811yescharacteristic
188C23×C3C2×M1611yescharacteristic
189C23×C3#13112yes—
190C3×D4—48no—
191C3×D4—48no—
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—24no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—84no—
212C2×C3×C4C22⋊C812yes—
213C2×C3×C4C22⋊C812yes—
214C2×C3×C4#9912yes—
215C2×C3×C4#10012yes—
216C2×C3×C4#10012yes—
217C2×C3×C4#12112yes—
218C2×C3×C4#12411yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C3⋊C8—44no—
221C3⋊C8—82no—
222C24—22no—
223C24—22no—
224C24—24no—
225C24—24no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24C2×C3⋊C814yes—
231C24C22×Dic311yescharacteristic
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—48no—
237C2×D4—48no—
238C2×D4—42no—
239C2×D4—42no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×C3⋊C812yes—
252C22×C4#20812yes—
253C22×C4#21711yescharacteristic
254C22×C4#21711yescharacteristic
255C22×C4#21711yescharacteristic
256C22×C4#21711yescharacteristic
257C22×C4D3×D412yes—
258C22×C4#23712yes—
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C2×C8—64no—
262C2×C8—122no—
263C2×C8—122no—
264C2×C8—122no—
265C2×C8—122no—
266C2×C8—122no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—44no—
273C22⋊C4—42no—
274C42—22no—
275C42—22no—
276C42—41no—
277C42—41no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—44no—
289C22×C3—88no—
290C22×C3—88no—
291C22×C3—82no—
292C22×C3—82no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3#39011yescharacteristic
296C22×C3C2×C22⋊C811yescharacteristic
297C22×C3#40811yescharacteristic
298C22×C3C2×#9911yescharacteristic
299C22×C3#41011yescharacteristic
300C22×C3C2×#10011yescharacteristic
301C22×C3C2×#10011yescharacteristic
302C22×C3C8×D412yes—
303C22×C3#43612yes—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—41no—
310C3×C4—82no—
311C3×C4#32314yescomplemented
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—24no—
317C23—28no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23C22×C3⋊C811yescharacteristic
334C23C2×#20811yescharacteristic
335C23D4×Dic312yes—
336C23#94812yes—
337C23C2×#21711yescharacteristic
338D4—48no—
339D4—48no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—84no—
360C2×C4#83012yes—
361C2×C4#83012yes—
362C2×C4#83312yes—
363C2×C4#83312yes—
364C2×C4#83412yes—
365C2×C4#93611yescharacteristic
366C2×C4#93812yes—
367C2×C4#94011yescharacteristic
368C8—124no—
369C8—242no—
370C2×C3—22no—
371C2×C3—22no—
372C2×C3—84no—
373C2×C3C2×#32311yescharacteristic
374C2×C3#146011yescharacteristic
375C2×C3#146911yescharacteristic
376C2×C3#147111yescharacteristic
377C2×C3#195411yescharacteristic
378C2×C3#196411yescharacteristic
379C2×C3#196511yescharacteristic
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—88no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22D4×C3⋊C812yes—
398C22#501212yes—
399C22C2×#83011yescharacteristic
400C22C2×#83311yescharacteristic
401C22C2×#83311yescharacteristic
402C22C2×#83411yescharacteristic
403C22#522111yescharacteristic
404C22#523411yescharacteristic
405C22#524011yescharacteristic
406C4—22no—
407C4—21no—
408C4—21no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—82no—
413C4#452514yescomplemented
414C3#964911yescharacteristic, complemented, Sylow
415C2—22no—
416C2—22no—
417C2—84no—
418C2C2×#452511yescharacteristic
419C2#7160111yescharacteristic
420C2#7161411yescharacteristic
421C2#7167611yescharacteristic
422C2#7168611yescharacteristic
423C2#7172011yescharacteristic
424C2#7176611yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC2×C3×#390C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C4
cosocleC23×C3×C4C23
hypercenterC2×#390D3
hypocenterC3#9649
LCSG ▷ C23×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. C4×#830C2semidirect
32. C22×C3×D4C8semidirect
56. #830D4semidirect
137. C22×D4C3⋊C8semidirect
311. C3×C4#323semidirect
413. C4#4525semidirect
414. C3#9649semidirect

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
C4×#8302
#96493