Finite Group Databasebeta

← #10692927

Subgroups — #10692927

Order 768.

1034 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2433C211yesmaximal, characteristic
2#84043C211yesmaximal, characteristic, complemented
3#72129C211yesmaximal, characteristic, complemented
4#71674C212yesmaximal, complemented
5#67616C212yesmaximal, complemented
6#9463—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#127C412yescomplemented
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C2×C3×#120—21no—
12C22×#217—21no—
13C3×#392—22no—
14C3×#390C412yescomplemented
15C3×C4×C22⋊C4C2211yescharacteristic
16#5247—21no—
17#5220—24no—
18C2×#833—22no—
19#5211C2212yes—
20#5207C2212yes—
21#4537—24no—
22#2433D311yescharacteristic
23#2308—31no—
24#1783—31no—
25#1510—32no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C25×C3D411yescharacteristic
29C22×C3×D4—21no—
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—44no—
35C22×C3×D4C2×C412yes—
36C22×C3×D4C2×C411yescharacteristic, complemented
37C22×C3×D4D412yes—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—22no—
40C3×#127—44no—
41C3×#121—22no—
42C3×#121—22no—
43C3×#120—44no—
44C3×#120—44no—
45C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2×C412yescomplemented
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—44no—
56C2×#208—22no—
57C3×#95—21no—
58#833—44no—
59C3⋊#95—22no—
60C3⋊#95D414yescomplemented
61#830—22no—
62#830—22no—
63C23×D4C2×D311yescharacteristic
64C2×#127Dic312yescomplemented
65C2×#121C2×D311yescharacteristic
66C2×#120—21no—
67C2×#120—21no—
68C22×C22⋊C4—61no—
69C2×#100—62no—
70#407—32no—
71#392—22no—
72#390Dic312yescomplemented
73#386—64no—
74#381—32no—
75#379—61no—
76C4×C22⋊C4C2×D311yescharacteristic
77#327—64no—
78C24×C3—21no—
79C24×C3—24no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—21no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—81no—
88C24×C3—81no—
89C24×C3C22⋊C412yes—
90C24×C3C22⋊C412yes—
91C24×C3C2×D411yescharacteristic
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—24no—
95C2×C3×D4—22no—
96C2×C3×D4—24no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C2×C3×D4—42no—
100C2×C3×D4—44no—
101C2×C3×D4—44no—
102C2×C3×D4—44no—
103C2×C3×D4—44no—
104C2×C3×D4—48no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—44no—
108C2×C3×D4—44no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—82no—
112C2×C3×D4—82no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—42no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4112yes—
120C22×Dic3—24no—
121C22×Dic3—44no—
122C22×Dic3—82no—
123C22×Dic3—82no—
124C3×C4⋊C4—22no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—44no—
127C3×C22⋊C4—44no—
128C3×C22⋊C4—42no—
129C3×C42—42no—
130#217—44no—
131#217—44no—
132#217—82no—
133#217—82no—
134#208—44no—
135C2×C3⋊C8—42no—
136C25#6711yescharacteristic
137C25#6711yescharacteristic
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—44no—
144C22×D4C2×Dic312yes—
145C22×D4C2×Dic311yescharacteristic, complemented
146C22×D4#6712yes—
147C23×C4—62no—
148C23×C4C22×D311yescharacteristic
149C2×M16—62no—
150#127—44no—
151#121—22no—
152#121—22no—
153#120—44no—
154#120—44no—
155C2×C4⋊C4C2×Dic311yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—124no—
165C2×C42C2×Dic312yescomplemented
166#100—124no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#95—21no—
170#95—34no—
171#95—62no—
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—24no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—42no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—42no—
198C23×C3—81no—
199C23×C3—81no—
200C23×C3—81no—
201C23×C3—81no—
202C23×C3—88no—
203C23×C3—88no—
204C23×C3#9914yes—
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3#12011yescharacteristic
207C23×C3#12311yescharacteristic
208C3×D4—48no—
209C3×D4—48no—
210C3×D4—82no—
211C3×D4—88no—
212C3×D4—88no—
213C3×D4—82no—
214C3×D4—82no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4C4×D412yes—
226C2×C3×C4#12112yes—
227C2×Dic3—44no—
228C2×Dic3—44no—
229C2×Dic3—44no—
230C2×Dic3—88no—
231C3⋊C8—82no—
232C24—21no—
233C24—24no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—81no—
242C24—81no—
243C24#21712yes—
244C24#21712yes—
245C24C2×#6711yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—48no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—64no—
271C22×C4—124no—
272C22×C4—242no—
273C22×C4—242no—
274C22×C4D3×D412yes—
275C22×C4#23712yes—
276C22×C4C22×Dic311yescharacteristic
277C22×C4C2×#6711yescharacteristic
278M16—124no—
279C2×C8—122no—
280C4⋊C4—22no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—124no—
286C22⋊C4—124no—
287C22⋊C4—242no—
288C22⋊C4—242no—
289C42—42no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—24no—
297C22×C3—21no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—84no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—84no—
314C22×C3—84no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3#38611yescharacteristic
319C22×C3C2×#9912yes—
320C22×C3#45312yes—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—44no—
324C3×C4—81no—
325Dic3—84no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—88no—
357C23—88no—
358C23#83414yes—
359C23#93511yescharacteristic
360C23#93711yescharacteristic
361C23C2×#21711yescharacteristic
362D4—48no—
363D4—48no—
364D4—82no—
365D4—88no—
366D4—88no—
367D4—82no—
368D4—82no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—124no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—248no—
383C2×C4D4×Dic312yes—
384C2×C4#93912yes—
385C8—242no—
386C2×C3—22no—
387C2×C3—22no—
388C2×C3—41no—
389C2×C3—44no—
390C2×C3—82no—
391C2×C3—161no—
392C2×C3#188711yescharacteristic
393C2×C3#189811yescharacteristic
394C2×C3#191111yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22—82no—
415C22—84no—
416C22—82no—
417C22—82no—
418C22—84no—
419C22—84no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#507712yes—
424C22C2×#83412yes—
425C22#522011yescharacteristic
426C4—22no—
427C4—22no—
428C4—44no—
429C4—81no—
430C4—244no—
431C3#946311yescharacteristic, complemented, Sylow
432C2—22no—
433C2—22no—
434C2—41no—
435C2—44no—
436C2—82no—
437C2—161no—
438C2#7167011yescharacteristic
439C2#7176311yescharacteristic
440C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2433C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2433D3
hypocenterC3#9463
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2433

Complements

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

NH ≅ G/Ntype
2. #84043C2semidirect
3. #72129C2semidirect
4. #71674C2semidirect
5. #67616C2semidirect
8. C2×C3×#127C4semidirect
14. C3×#390C4semidirect
36. C22×C3×D4C2×C4semidirect
45. C2×C3×C4⋊C4C2×C4semidirect
50. C2×C3×C42C2×C4semidirect
60. C3⋊#95D4semidirect
64. C2×#127Dic3semidirect
72. #390Dic3semidirect
145. C22×D4C2×Dic3semidirect
155. C2×C4⋊C4C2×Dic3semidirect
165. C2×C42C2×Dic3semidirect
431. C3#9463semidirect

Maximal subgroups

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

subgroupindex
C3×#24332
#840432
#721292
#716742
#676162
#94633