Finite Group Databasebeta

← #10692631

Subgroups — #10692631

Order 768.

840 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 97 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#73560C211yesmaximal, characteristic, complemented
3#69977C211yesmaximal, characteristic, complemented
4#69967C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9409—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#886—21no—
12C2×#882—21no—
13C2×D3×C22⋊C4—21no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19Dic3×C22⋊C4—21no—
20#4939—22no—
21C4×(C2×D3)⋊C4—22no—
22#4677C412yescomplemented
23#4671C412yescomplemented
24#4666—22no—
25#4470—22no—
26#4468—22no—
27#4451—22no—
28#2539—31no—
29#2273—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, complemented
41C22×C4×D3—22no—
42C2×C3×M16—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42—21no—
47C2×#217—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4C2×C412yes—
61C2×(C2×D3)⋊C4D412yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3C2×C411yescharacteristic, complemented
64C2×C4×Dic3—22no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67#886—44no—
68#882—44no—
69D3×C22⋊C4—28no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95C2×C412yescomplemented
73C3×#95D412yescomplemented
74C3⋊#95—22no—
75C3⋊#95D412yescomplemented
76#830—21no—
77#830—21no—
78C2×#123—61no—
79C2×#121—31no—
80C2×#120—61no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—61no—
83#407—31no—
84#407C2×D311yescharacteristic
85#399—32no—
86#392—32no—
87#390—62no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4—62no—
90C4×C22⋊C4—61no—
91C4×C22⋊C4—62no—
92C4×C22⋊C4C2×D311yescharacteristic
93C4×C22⋊C4C2×D311yescharacteristic
94#327—62no—
95#327—62no—
96#323—62no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—24no—
101C23×D3—21no—
102C23×D3—42no—
103C23×D3—81no—
104C23×D3C22⋊C412yes—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—81no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—42no—
115C2×#67—42no—
116C2×#67—41no—
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—42no—
123C22×Dic3C22⋊C412yes—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—44no—
127C2×D12—44no—
128C2×D12—82no—
129C2×D12—81no—
130C2×C4×D3—28no—
131C2×C4×D3—82no—
132C2×C4×D3—82no—
133C3×M16—81no—
134C2×C3×C8—41no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C42—81no—
138C3×C42—81no—
139#217—24no—
140(C2×D3)⋊C4—28no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—44no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—44no—
152C4×Dic3—81no—
153C4×Dic3—82no—
154#208—81no—
155C2×C3⋊C8—41no—
156C25—31no—
157C22×D4—31no—
158C22×D4—62no—
159C22×D4—61no—
160C23×C4—31no—
161C23×C4—62no—
162C23×C4C22×D311yescharacteristic
163C2×M16—21no—
164C2×M16—61no—
165#123—124no—
166#121—61no—
167#121—61no—
168#121—61no—
169#121—61no—
170#120—124no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—62no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—32no—
176C2×C22⋊C4—32no—
177C2×C22⋊C4—68no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—61no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—62no—
190C2×C42—61no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195#95—32no—
196#95—62no—
197#95C4×D312yescomplemented
198#95D1212yescomplemented
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3C2×C22⋊C411yescharacteristic
203C23×C3#12011yescharacteristic
204C23×C3#12311yescharacteristic
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—24no—
209C22×D3—22no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—42no—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—88no—
220C22×D3—81no—
221C22×D3—81no—
222C22×D3#10414yes—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239#67—82no—
240#67—82no—
241C2×Dic3—24no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—81no—
251D12—81no—
252D12—81no—
253D12—88no—
254C4×D3—88no—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—32no—
258C24—64no—
259C24—61no—
260C24—61no—
261C24—61no—
262C24—122no—
263C24—241no—
264C24D3×D411yescharacteristic
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—124no—
268C2×D4—124no—
269C2×D4—122no—
270C2×D4—122no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—242no—
274C2×D4—241no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—32no—
278C22×C4—68no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—81no—
284C22×C4—122no—
285C22×C4—242no—
286C22×C4—242no—
287C22×C4C2×C4×D311yescharacteristic
288C22×C4C2×D1211yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23911yescharacteristic
293M16—81no—
294M16—241no—
295C2×C8—41no—
296C2×C8—121no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—124no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—68no—
303C22⋊C4—64no—
304C22⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—124no—
309C22⋊C4—124no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—81no—
313C42—81no—
314C42—241no—
315C42—242no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—44no—
324C22×C3#38611yescharacteristic
325C22×C3C2×#10412yes—
326C22×C3#45312yes—
327C2×D3—24no—
328C2×D3—44no—
329C2×D3—44no—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—44no—
333C2×D3—82no—
334C2×D3—84no—
335C2×D3—82no—
336C2×D3—82no—
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—44no—
345C3×C4—44no—
346Dic3—44no—
347Dic3—42no—
348Dic3—81no—
349C23—22no—
350C23—22no—
351C23—34no—
352C23—41no—
353C23—62no—
354C23—62no—
355C23—64no—
356C23—62no—
357C23—62no—
358C23—122no—
359C23—122no—
360C23—122no—
361C23—121no—
362C23—121no—
363C23—122no—
364C23—122no—
365C23—121no—
366C23—122no—
367C23—241no—
368C23—241no—
369C23—248no—
370C23D3×C22⋊C411yescharacteristic
371C23#88211yescharacteristic
372C23#89411yescharacteristic
373D4—241no—
374D4—241no—
375D4—248no—
376D4—242no—
377D4—242no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—248no—
400C2×C4—241no—
401C2×C4C4×D1211yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#89311yescharacteristic
404C2×C4#89511yescharacteristic
405C8—81no—
406C8—241no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—41no—
410C2×C3#188711yescharacteristic
411C2×C3#189611yescharacteristic
412C2×C3#191311yescharacteristic
413D3—44no—
414D3—161no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—64no—
424C22—124no—
425C22—124no—
426C22—124no—
427C22—122no—
428C22—122no—
429C22—242no—
430C22—242no—
431C22—242no—
432C22—244no—
433C22—241no—
434C22—241no—
435C22—241no—
436C22#466511yescharacteristic
437C22D3×#10412yes—
438C22#482212yes—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—44no—
444C4—44no—
445C4—124no—
446C4—122no—
447C4—241no—
448C3#940911yescharacteristic, complemented, Sylow
449C2—22no—
450C2—22no—
451C2—41no—
452C2—124no—
453C2—481no—
454C2#6875611yescharacteristic
455C2#6876011yescharacteristic
456C2#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#9409
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. #73560C2semidirect
3. #69977C2semidirect
4. #69967C2semidirect
5. #69444C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4677C4semidirect
23. #4671C4semidirect
34. #1297D3semidirect
40. C22×D12C2×C4semidirect
63. C2×C4⋊Dic3C2×C4semidirect
72. C3×#95C2×C4semidirect
73. C3×#95D4semidirect
75. C3⋊#95D4semidirect
197. #95C4×D3semidirect
198. #95D12semidirect
448. C3#9409semidirect

Maximal subgroups

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

subgroupindex
#735972
#735602
#699772
#699672
#694442
C3×#12972
#676182
#94093