Finite Group Databasebeta

← #10690534

Subgroups — #10690534

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic, complemented
5#67619C212yesmaximal, complemented
6#9074—31nomaximal, Sylow
7C2×C3×#121C411yescharacteristic
8C2×C3×C4×D4—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C3×#390C411yescharacteristic
12C3×#389C411yescharacteristic
13C3×#388—21no—
14C3×#386—21no—
15C3×#380C411yescharacteristic
16C3×C4×C4⋊C4C2211yescharacteristic
17C3×C4×C22⋊C4—21no—
18C2×#830—21no—
19C2×#830—21no—
20#4925—24no—
21C2×C4⋊(C3⋊C8)—21no—
22C2×C4⋊(C3⋊C8)C2212yes—
23C2×C4×C3⋊C8C2212yes—
24D4⋊(C3⋊C8)—24no—
25#4524—21no—
26C2×#431—31no—
27C4×#121D311yescharacteristic
28#1797—31no—
29C2×D4⋊C8—32no—
30#1279—32no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C3×#121—22no—
38C3×#121C814yescomplemented
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C3×C4×D4—44no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42—22no—
52C2×C3×C42—22no—
53C2×C3×C42—21no—
54C2×C3×C42C2×C411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C22×C3⋊C8—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63#830—24no—
64#830—42no—
65C4⋊(C3⋊C8)—24no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—22no—
68C4×C3⋊C8—22no—
69C4×C3⋊C8D414yescomplemented
70C2×#121Dic311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×D311yescharacteristic
73C22×C42C2×D311yescharacteristic
74#431—64no—
75C2×C4⋊C8—32no—
76C2×C4⋊C8—61no—
77C2×C22⋊C8—61no—
78C2×C22⋊C8—61no—
79C2×C4×C8—32no—
80#390Dic311yescharacteristic
81#389Dic311yescharacteristic
82#388—21no—
83#386—21no—
84#380Dic311yescharacteristic
85C4×C4⋊C4C2×D311yescharacteristic
86C4×C22⋊C4—21no—
87#336—61no—
88D4⋊C8—64no—
89C24×C3—21no—
90C24×C3C22⋊C411yescharacteristic
91C2×C3×D4—24no—
92C2×C3×D4—22no—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—42no—
96C2×C3×D4—42no—
97C2×C3×D4C2×C812yes—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C2×C812yes—
119C22×C3×C4M1612yes—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4C2×C812yes—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—22no—
135C3×C42—21no—
136C3×C42—22no—
137C3×C42—21no—
138C3×C42—42no—
139C3×C42C2×D412yes—
140C3×C42#4112yes—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—42no—
143C2×C3⋊C8—42no—
144C2×C3⋊C8—42no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—42no—
147C22×D4—21no—
148C22×D4C2×Dic311yescharacteristic
149C23×C4—21no—
150C23×C4C2×Dic311yescharacteristic
151C23×C4#6711yescharacteristic
152C23×C4#6711yescharacteristic
153C22×C8—62no—
154C22×C8—62no—
155#121—22no—
156#121C3⋊C814yescomplemented
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4C2×Dic311yescharacteristic
162C2×C4⋊C4C2×Dic311yescharacteristic
163C2×C4⋊C4C2×Dic311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42C2×Dic311yescharacteristic
173C2×C42#6711yescharacteristic
174C2×C42#6711yescharacteristic
175C2×C42C22×D311yescharacteristic
176C4⋊C8—64no—
177C4⋊C8—62no—
178C4⋊C8—62no—
179C22⋊C8—64no—
180C22⋊C8—122no—
181C4×C8—34no—
182C4×C8—62no—
183#95—21no—
184#95—21no—
185#95—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—22no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3C22⋊C812yes—
200C23×C3C2×C22⋊C411yescharacteristic
201C3×D4—24no—
202C3×D4—48no—
203C3×D4—42no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4C22⋊C812yes—
232C2×C3×C4C4×D412yes—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12112yes—
235C2×C3×C4#12311yescharacteristic
236C2×C3×C4C22×C811yescharacteristic
237C2×C3×C4C2×M1611yescharacteristic
238C2×C3×C4#13112yes—
239C2×C3×C4AGL(1,ℤ/8)12yes—
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4#13711yescharacteristic
242C3⋊C8—44no—
243C3⋊C8—82no—
244C24—21no—
245C24#21711yescharacteristic
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4C2×C3⋊C812yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×C3⋊C812yes—
271C22×C4#20812yes—
272C22×C4#21711yescharacteristic
273C22×C4#21711yescharacteristic
274C22×C4#21711yescharacteristic
275C22×C4C22×Dic311yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C2×C8—64no—
279C2×C8—122no—
280C2×C8—122no—
281C2×C8—122no—
282C2×C8—122no—
283C2×C8—122no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4C2×C3⋊C812yes—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C42—22no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—42no—
300C42D3×D412yes—
301C42#23712yes—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#38611yescharacteristic
317C22×C3C2×C22⋊C811yescharacteristic
318C22×C3#40811yescharacteristic
319C22×C3#41711yescharacteristic
320C22×C3#41811yescharacteristic
321C22×C3#42111yescharacteristic
322C22×C3#42111yescharacteristic
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—42no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4C8×D412yes—
332C3×C4#43512yes—
333C3×C4#46812yes—
334C3×C4#46912yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23#83012yes—
349C23C2×#21711yescharacteristic
350D4—24no—
351D4—48no—
352D4—42no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#83012yes—
381C2×C4C22×C3⋊C811yescharacteristic
382C2×C4C2×#20811yescharacteristic
383C2×C4#93212yes—
384C2×C4D4×Dic312yes—
385C2×C4#93511yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#93912yes—
388C2×C4#94812yes—
389C2×C4#94911yescharacteristic
390C2×C4#95111yescharacteristic
391C8—124no—
392C8—242no—
393C2×C3—42no—
394C2×C3—82no—
395C2×C3#147711yescharacteristic
396C2×C3#147811yescharacteristic
397C2×C3#149511yescharacteristic
398C2×C3#149611yescharacteristic
399C2×C3#186911yescharacteristic
400C2×C3#189511yescharacteristic
401C2×C3#189711yescharacteristic
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22C2×#83011yescharacteristic
417C22#521511yescharacteristic
418C22#522011yescharacteristic
419C22#523211yescharacteristic
420C22#523411yescharacteristic
421C22#523611yescharacteristic
422C22#523611yescharacteristic
423C4—21no—
424C4—21no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—42no—
429C4—81no—
430C4—81no—
431C4D4×C3⋊C812yes—
432C4#501112yes—
433C4#503912yes—
434C4#504212yes—
435C3#907411yescharacteristic, complemented, Sylow
436C2—42no—
437C2—82no—
438C2#7159711yescharacteristic
439C2#7161011yescharacteristic
440C2#7163911yescharacteristic
441C2#7168211yescharacteristic
442C2#7171211yescharacteristic
443C2#7175311yescharacteristic
444C2#7180311yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×C4×#121C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C4
cosocleC2×C3×C42C23
hypercenterC4×#121D3
hypocenterC3#9074
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4925C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #71530C2semidirect
5. #67619C2semidirect
38. C3×#121C8semidirect
69. C4×C3⋊C8D4semidirect
156. #121C3⋊C8semidirect
435. C3#9074semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C2×#49252
C2×D4⋊(C3⋊C8)2
#715302
#676192
#90743