Finite Group Databasebeta

← #10707841

Subgroups — #10707841

Order 768.

442 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 55 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#76845C211yesmaximal, characteristic, complemented
2C3×#1601C211yesmaximal, characteristic
3#71849C211yesmaximal, characteristic
4#71843C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic, complemented
6#67450C211yesmaximal, characteristic, complemented
7#67445C211yesmaximal, characteristic, complemented
8#12193—31nomaximal, Sylow
9C3×#534C2211yescharacteristic
10C2×#935—21no—
11C3×#448—21no—
12C3×#393C2211yescharacteristic
13#5229—21no—
14C2×#834—21no—
15#5033—21no—
16#4953C2211yescharacteristic
17#4950—21no—
18#4944—21no—
19#4943C2211yescharacteristic
20#4661—21no—
21#4660—21no—
22#4651—21no—
23C3×#327—21no—
24C3×#323C2211yescharacteristic
25#4537—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2764—31no—
29#2022—31no—
30#1607—31no—
31#1601D311yescharacteristic
32#1351—31no—
33#1344—31no—
34#1317—31no—
35C22×C3×Q8D411yescharacteristic
36C22×C3×D4—21no—
37C23×Dic3D411yescharacteristic
38C2×C3×SD16—41no—
39C3×#123—41no—
40C3×#121—21no—
41C3×#121—21no—
42C3×#120—41no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—41no—
48C2×#217D411yescharacteristic
49#935—42no—
50#935—42no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—41no—
56C2×C4×Dic3—21no—
57C3×#103—41no—
58C3×C22⋊C8—21no—
59C3×#95—21no—
60C3×#95D411yescharacteristic
61#834—42no—
62#832—41no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71#830—21no—
72#807—41no—
73#534C2×D311yescharacteristic
74C2×#123—61no—
75#480—61no—
76#448—21no—
77C2×#99—61no—
78#400—61no—
79#398—61no—
80#397—31no—
81#393C2×D311yescharacteristic
82#391—61no—
83#388—61no—
84#385—61no—
85#383—61no—
86#381—31no—
87#342—31no—
88#327—21no—
89#327—61no—
90#323—31no—
91#323C2×D311yescharacteristic
92C24×C3—21no—
93C24×C3C2×D411yescharacteristic
94C2×C3×Q8—42no—
95C2×C3×Q8—41no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—41no—
118C3×SD16—82no—
119C2×C3×C8—41no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126#217—22no—
127#217—44no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—81no—
134C4⋊Dic3—81no—
135Dic3⋊C4—44no—
136Dic3⋊C4—81no—
137C4×Dic3—42no—
138C2×C3⋊C8—41no—
139C22×Q8#6711yescharacteristic
140C22×D4—21no—
141C23×C4—31no—
142C2×SD16—41no—
143#123—41no—
144#123—122no—
145#123—122no—
146#121—21no—
147#121—21no—
148#120—41no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—121no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—121no—
159C2×C22⋊C4C22×D311yescharacteristic
160C2×C42—61no—
161#107—121no—
162#103—41no—
163#102—121no—
164#99—122no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167#95—21no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95#6711yescharacteristic
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3#12011yescharacteristic
187C23×C3#13511yescharacteristic
188C23×C3AGL(1,ℤ/8)11yescharacteristic
189C3×Q8—42no—
190C3×Q8—81no—
191C3×D4—81no—
192C3×D4—81no—
193C3×D4—81no—
194C3×D4—81no—
195C3×D4—81no—
196C3×D4—81no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C3×C8—81no—
228C3⋊C8—81no—
229C24—21no—
230C24D3×D411yescharacteristic
231C2×Q8—42no—
232C2×Q8—41no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4C2×#6711yescharacteristic
255SD16—82no—
256C2×C8—41no—
257C2×C8—121no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—121no—
261C4⋊C4—124no—
262C4⋊C4—241no—
263C4⋊C4—241no—
264C4⋊C4—241no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—62no—
270C22⋊C4—124no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42—122no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3—82no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—161no—
290C22×C3#44911yescharacteristic
291C22×C3#45311yescharacteristic
292C22×C3B(4,𝔽2)11yescharacteristic
293C3×C4—41no—
294C3×C4—42no—
295C3×C4—41no—
296C3×C4—81no—
297C3×C4—81no—
298Dic3—81no—
299Dic3—81no—
300Dic3—82no—
301Dic3—81no—
302Dic3—81no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—81no—
311C23—81no—
312C23#91411yescharacteristic
313C23#91611yescharacteristic
314C23#93711yescharacteristic
315Q8—42no—
316Q8—81no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4—241no—
353C8—81no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—161no—
360C2×C3#160011yescharacteristic
361C2×C3#219711yescharacteristic
362C2×C3#220111yescharacteristic
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—161no—
377C22#503211yescharacteristic
378C22#517211yescharacteristic
379C22#519811yescharacteristic
380C4—41no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—241no—
386C4—241no—
387C4—242no—
388C4—241no—
389C4—241no—
390C3#1219311yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2#7097711yescharacteristic
397C2#7099211yescharacteristic
398C2#7099811yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1601C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#534C22
hypercenter#1601D3
hypocenterC3#12193
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1601

Complements

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

NH ≅ G/Ntype
1. #76845C2semidirect
5. #67646C2semidirect
6. #67450C2semidirect
7. #67445C2semidirect
390. C3#12193semidirect

Maximal subgroups

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

subgroupindex
#768452
C3×#16012
#718492
#718432
#676462
#674502
#674452
#121933