Finite Group Databasebeta

← #10692382

Subgroups — #10692382

Order 768.

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

normal subgroups only

This group has 542 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#78624C211yesmaximal, characteristic
2#73882C211yesmaximal, characteristic, complemented
3#69977C211yesmaximal, characteristic
4#69975C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#9366—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C4×#217C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5131C412yescomplemented
17#5121C412yescomplemented
18Dic3×C4⋊C4—21no—
19#4972—21no—
20C2×#806—22no—
21#4944—22no—
22#4737—22no—
23#4736—21no—
24#4735—21no—
25#4651—21no—
26#4649—21no—
27#4555—22no—
28#4470—21no—
29#4470—21no—
30#3094—31no—
31#2347—31no—
32#1964—31no—
33#1826—31no—
34#1507—31no—
35#1305D311yescharacteristic, complemented
36#1297—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C22×D12C2×C411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×M16C2×C412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4D412yes—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51C2×(C2×D3)⋊C4—21no—
52#926—21no—
53#926—21no—
54#926C2×C412yes—
55#926D412yes—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—22no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—22no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#821—22no—
75#821—22no—
76#818—22no—
77#817—22no—
78#806—44no—
79C2×#121—31no—
80C2×#117—31no—
81#482—62no—
82#481—61no—
83#480—61no—
84#428—32no—
85#418—32no—
86C2×#99—62no—
87#407C2×D311yescharacteristic
88#407C2×D311yescharacteristic
89#389—61no—
90#388—62no—
91#383—61no—
92#380C2×D311yescharacteristic
93C4×C4⋊C4—61no—
94C4×C4⋊C4—61no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4—31no—
97#339—62no—
98#327—61no—
99#327—61no—
100C24×C3C2×D411yescharacteristic
101C23×D3—21no—
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—42no—
111C2×#67—42no—
112C2×#67—41no—
113C2×#67—41no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C3×M16—42no—
124C3×M16—42no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—42no—
128C3×C22⋊C4—42no—
129#217—22no—
130#217—22no—
131#217—22no—
132#217—42no—
133#217—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3C22⋊C412yes—
141C4⋊Dic3C22⋊C412yes—
142Dic3⋊C4—44no—
143C4×Dic3—22no—
144C4×Dic3—22no—
145C4×Dic3—42no—
146C4×Dic3—42no—
147C4×Dic3—82no—
148C22×D4—31no—
149C22×D4—61no—
150C23×C4C22×D311yescharacteristic
151C2×M16—21no—
152C2×M16C4×D312yes—
153#121—61no—
154#121—61no—
155#121—61no—
156#121—61no—
157#117—32no—
158#117—32no—
159#117—61no—
160#117—61no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—62no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4D1212yes—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—62no—
175#107—62no—
176#106—62no—
177#102—62no—
178#102—62no—
179#99—124no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184#95—21no—
185#95—31no—
186#95—31no—
187#95—62no—
188#95—62no—
189#95—61no—
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—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214#67—82no—
215#67—81no—
216#67—81no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233D12—42no—
234D12—81no—
235C3×C8—42no—
236C3×C8—81no—
237C24—61no—
238C24C2×D1211yescharacteristic
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—121no—
245C2×D4—121no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4#23511yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4#23911yescharacteristic
260M16—42no—
261M16—42no—
262C2×C8—22no—
263C2×C8—41no—
264C4⋊C4—32no—
265C4⋊C4—32no—
266C4⋊C4—42no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—124no—
272C22⋊C4—42no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—122no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—62no—
281C42—62no—
282C42—122no—
283C42—122no—
284C42—242no—
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—82no—
306C3×C4#41012yes—
307Dic3—42no—
308Dic3—42no—
309Dic3—42no—
310Dic3—81no—
311Dic3—82no—
312Dic3—81no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—122no—
318C23—121no—
319C23—241no—
320C23—241no—
321C23C4×D1211yescharacteristic
322C23#88211yescharacteristic
323C23#89511yescharacteristic
324D4—122no—
325D4—241no—
326D4—242no—
327D4—241no—
328D4—241no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—121no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4D3×C22⋊C411yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4#89311yescharacteristic
357C2×C4#89411yescharacteristic
358C8—42no—
359C8—81no—
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#466511yescharacteristic
383C22#474711yescharacteristic
384C22#475311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—122no—
390C4—122no—
391C4—122no—
392C4—241no—
393C4—242no—
394C4—241no—
395C4#474012yes—
396C3#936611yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—481no—
402C2#6895411yescharacteristic
403C2#6900511yescharacteristic
404C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9366
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #73882C2semidirect
4. #69975C2semidirect
6. C3×#1305C2semidirect
7. #67683C2semidirect
16. #5131C4semidirect
17. #5121C4semidirect
35. #1305D3semidirect
396. C3#9366semidirect

Maximal subgroups

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

subgroupindex
#786242
#738822
#699772
#699752
#695282
C3×#13052
#676832
#93663