Finite Group Databasebeta

← #10692651

Subgroups — #10692651

Order 768.

720 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#73597C211yesmaximal, characteristic
2#73567C211yesmaximal, characteristic, complemented
3#69977C211yesmaximal, characteristic, complemented
4#69967C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9413—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#886—21no—
12C2×#882—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#4955—21no—
19#4942—21no—
20#4940—22no—
21#4939—22no—
22#4677C412yescomplemented
23#4671C412yescomplemented
24#4669—22no—
25#4470—22no—
26#4468—22no—
27#4451—22no—
28#2539—31no—
29#2312—31no—
30#1507—31no—
31#1507—31no—
32#1462—31no—
33#1297—31no—
34#1297D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C22×#67—21no—
38C23×Dic3D411yescharacteristic
39C22×D12—22no—
40C22×D12C2×C411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42—21no—
46C2×#217—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4C2×C412yes—
61C2×(C2×D3)⋊C4D412yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67#886—44no—
68#882—44no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95C2×C412yes—
72C3×#95D412yescomplemented
73C3⋊#95—22no—
74C3⋊#95D412yescomplemented
75#830—21no—
76#830—21no—
77C2×#123—61no—
78C2×#121—31no—
79C2×#120—61no—
80C22×C22⋊C4—31no—
81#407—31no—
82#407C2×D311yescharacteristic
83#399—32no—
84#392—32no—
85#390—62no—
86#388—62no—
87#388—62no—
88#380—61no—
89#379—61no—
90C4×C22⋊C4—31no—
91C4×C22⋊C4C2×D311yescharacteristic
92C4×C22⋊C4C2×D311yescharacteristic
93#327—62no—
94#327—62no—
95#323—62no—
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—42no—
101C23×D3—42no—
102C23×D3—81no—
103C23×D3C22⋊C412yes—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—81no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—42no—
114C2×#67—42no—
115C2×#67—41no—
116C2×#67—41no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C22×Dic3—42no—
122C22×Dic3C22⋊C412yes—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—44no—
126C2×D12—44no—
127C2×D12—82no—
128C2×D12—81no—
129C3×M16—81no—
130C2×C3×C8—41no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—42no—
133C3×C42—81no—
134C3×C42—81no—
135#217—42no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—44no—
141(C2×D3)⋊C4—44no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—82no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—44no—
149Dic3⋊C4—82no—
150C4×Dic3—81no—
151#208—81no—
152C2×C3⋊C8—41no—
153C25—31no—
154C22×D4—31no—
155C22×D4—62no—
156C22×D4—61no—
157C23×C4—31no—
158C23×C4C22×D311yescharacteristic
159C2×M16—21no—
160C2×M16—61no—
161#123—124no—
162#121—61no—
163#121—61no—
164#121—61no—
165#121—61no—
166#120—124no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—62no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—32no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—61no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191#95—32no—
192#95—62no—
193#95C4×D312yes—
194#95D1212yescomplemented
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—41no—
198C23×C3C2×C22⋊C411yescharacteristic
199C23×C3#12011yescharacteristic
200C23×C3#12311yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—42no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—42no—
213C22×D3—41no—
214C22×D3—42no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3—88no—
218C2×C3×C4—24no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234#67—82no—
235#67—82no—
236C2×Dic3—22no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—81no—
246D12—81no—
247D12—81no—
248D12—88no—
249C3×C8—81no—
250C3⋊C8—81no—
251C24—32no—
252C24—61no—
253C24—61no—
254C24—61no—
255C24—122no—
256C24—122no—
257C24—241no—
258C24D3×D411yescharacteristic
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—124no—
262C2×D4—124no—
263C2×D4—122no—
264C2×D4—122no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—242no—
268C2×D4—241no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—32no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—81no—
277C22×C4—122no—
278C22×C4C2×C4×D311yescharacteristic
279C22×C4C2×D1211yescharacteristic
280C22×C4#23511yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4#23911yescharacteristic
284M16—81no—
285M16—241no—
286C2×C8—41no—
287C2×C8—121no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—124no—
291C4⋊C4—242no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C22⋊C4—122no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—242no—
304C22⋊C4—242no—
305C42—81no—
306C42—81no—
307C42—241no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3#38611yescharacteristic
317C22×C3#42112yes—
318C22×C3#45312yes—
319C2×D3—44no—
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C2×D3—84no—
324C2×D3—84no—
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—44no—
336C3×C4—82no—
337Dic3—42no—
338Dic3—82no—
339Dic3—81no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—62no—
344C23—62no—
345C23—62no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—121no—
352C23—121no—
353C23—122no—
354C23—122no—
355C23—121no—
356C23—122no—
357C23—241no—
358C23—241no—
359C23—248no—
360C23D3×C22⋊C411yescharacteristic
361C23#88211yescharacteristic
362C23#89411yescharacteristic
363D4—241no—
364D4—241no—
365D4—248no—
366D4—242no—
367D4—242no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—62no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4C4×D1211yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4#89311yescharacteristic
393C2×C4#89511yescharacteristic
394C8—81no—
395C8—241no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—41no—
399C2×C3#188711yescharacteristic
400C2×C3#189711yescharacteristic
401C2×C3#191311yescharacteristic
402D3—82no—
403D3—161no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—124no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—242no—
417C22—244no—
418C22—242no—
419C22—242no—
420C22—244no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22#466511yescharacteristic
425C22#482112yes—
426C22#482212yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—82no—
433C4—122no—
434C4—242no—
435C4—241no—
436C3#941311yescharacteristic, complemented, Sylow
437C2—22no—
438C2—22no—
439C2—41no—
440C2—242no—
441C2—481no—
442C2#6875711yescharacteristic
443C2#6876011yescharacteristic
444C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radical#73597C2
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9413
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
2. #73567C2semidirect
3. #69977C2semidirect
4. #69967C2semidirect
5. #69444C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4677C4semidirect
23. #4671C4semidirect
34. #1297D3semidirect
72. C3×#95D4semidirect
74. C3⋊#95D4semidirect
194. #95D12semidirect
436. C3#9413semidirect

Maximal subgroups

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

subgroupindex
#735972
#735672
#699772
#699672
#694442
C3×#12972
#676182
#94133