Finite Group Databasebeta

← #10692383

Subgroups — #10692383

Order 768.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2347C211yesmaximal, characteristic, complemented
2#82217C211yesmaximal, characteristic
3#71590C211yesmaximal, characteristic, complemented
4#69528C211yesmaximal, characteristic
5#69444C211yesmaximal, characteristic
6#67626C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9366—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#389—21no—
12C3×#388—22no—
13C3×#383—21no—
14C3×#380C2211yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20#5207C2211yescharacteristic
21#5003—22no—
22#5001C412yescomplemented
23#4995C412yescomplemented
24#4968—21no—
25#4967—21no—
26C2×#806—22no—
27#4532—22no—
28#4451—21no—
29#4451—21no—
30#3094—31no—
31#2347D311yescharacteristic, complemented
32#1964—31no—
33#1826—31no—
34#1507—31no—
35#1305—31no—
36#1297—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C22×D12C2×C411yescharacteristic
40C3×#117—21no—
41C3×#117—21no—
42C3×#117C2×C412yes—
43C3×#117D412yes—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4D412yes—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42—22no—
54C2×#217—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C2×#208C2×C412yes—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69C3⋊#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#809—22no—
75#809—22no—
76#808—22no—
77#807—22no—
78#806—44no—
79C2×#121—31no—
80C2×#117C2×D311yescharacteristic
81#482—62no—
82#481—61no—
83#480—61no—
84#428—32no—
85#418—32no—
86C2×#99—62no—
87#407—31no—
88#407—31no—
89#389—21no—
90#388—22no—
91#383—21no—
92#380C2×D311yescharacteristic
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4C2×D311yescharacteristic
97#339—62no—
98#327—61no—
99#327—61no—
100C24×C3C2×D411yescharacteristic
101C23×D3—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
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—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—44no—
127C3×C4⋊C4C22⋊C412yes—
128C3×C4⋊C4C22⋊C412yes—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—42no—
133C3×C42—22no—
134C3×C42—22no—
135C3×C42—42no—
136C3×C42—82no—
137#217—22no—
138#217—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4×Dic3—42no—
144#208—42no—
145#208—42no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×D4—31no—
149C22×D4—61no—
150C23×C4C22×D311yescharacteristic
151C2×M16—32no—
152C2×M16—61no—
153#121—61no—
154#121—61no—
155#121—61no—
156#121—61no—
157#117—21no—
158#117—21no—
159#117C4×D312yes—
160#117D1212yes—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4#6711yescharacteristic
165C2×C4⋊C4#6711yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4#6712yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—61no—
175#107—62no—
176#106—62no—
177#102—62no—
178#102—62no—
179#99—124no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#95—22no—
185#95—22no—
186#95—21no—
187#95—61no—
188#95#6711yescharacteristic
189#95#6711yescharacteristic
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3C4×D411yescharacteristic
195C23×C3#12011yescharacteristic
196C23×C3#12111yescharacteristic
197C22×D3—41no—
198C22×D3—42no—
199C22×D3—81no—
200C22×D3—81no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225#67—82no—
226#67—81no—
227#67—81no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233D12—42no—
234D12—81no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—61no—
238C24C2×#6711yescharacteristic
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—121no—
245C2×D4—121no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4C2×D1211yescharacteristic
256C22×C4#23511yescharacteristic
257C22×C4#23511yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260M16—122no—
261M16—122no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4(C2×D3)⋊C412yes—
271C4⋊C4(C2×D3)⋊C412yes—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—62no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—22no—
281C42—22no—
282C42—42no—
283C42—82no—
284C42—122no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#38611yescharacteristic
296C22×C3#45311yescharacteristic
297C22×C3#45411yescharacteristic
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—162no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—42no—
308C3×C4—82no—
309C3×C4—81no—
310C3×C4#41012yes—
311Dic3—82no—
312Dic3—81no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—122no—
318C23—121no—
319C23—241no—
320C23—241no—
321C23C4×#6711yescharacteristic
322C23#93011yescharacteristic
323C23#95311yescharacteristic
324D4—122no—
325D4—241no—
326D4—242no—
327D4—241no—
328D4—241no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4C2×(C2×D3)⋊C411yescharacteristic
355C2×C4C4×#6711yescharacteristic
356C2×C4#92911yescharacteristic
357C2×C4#93011yescharacteristic
358C8—122no—
359C8—241no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3#188311yescharacteristic
365C2×C3#189911yescharacteristic
366C2×C3#190611yescharacteristic
367D3—161no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—482no—
382C22#521311yescharacteristic
383C22#524111yescharacteristic
384C22#524511yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—82no—
392C4—81no—
393C4—242no—
394C4—241no—
395C4#500612yes—
396C3#936611yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—481no—
402C2#7159511yescharacteristic
403C2#7164511yescharacteristic
404C2#7166711yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2347C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2347D3
hypocenterC3#9366
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
1. C3×#2347C2semidirect
3. #71590C2semidirect
6. #67626C2semidirect
7. #67618C2semidirect
22. #5001C4semidirect
23. #4995C4semidirect
31. #2347D3semidirect
396. C3#9366semidirect

Maximal subgroups

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

subgroupindex
C3×#23472
#822172
#715902
#695282
#694442
#676262
#676182
#93663