Finite Group Databasebeta

← #10707811

Subgroups — #10707811

Order 768.

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

normal subgroups only

This group has 432 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#77617C211yesmaximal, characteristic, complemented
2C3×#1607C211yesmaximal, characteristic
3#71849C211yesmaximal, characteristic
4#71844C211yesmaximal, characteristic
5#67649C211yesmaximal, characteristic, complemented
6#67451C211yesmaximal, characteristic, complemented
7#67445C211yesmaximal, characteristic, complemented
8#12189—31nomaximal, Sylow
9C3×#534C2211yescharacteristic
10C2×#935—21no—
11C3×#480—21no—
12C3×#397C2211yescharacteristic
13C2×#834—21no—
14#5034—21no—
15#4976—21no—
16#4951—21no—
17#4949C2211yescharacteristic
18#4946—21no—
19#4943C2211yescharacteristic
20#4660—21no—
21#4660—21no—
22#4653—21no—
23C3×#327—21no—
24C3×#323C2211yescharacteristic
25#4537—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2771—31no—
29#2022—31no—
30#1613—31no—
31#1607D311yescharacteristic
32#1352—31no—
33#1344—31no—
34#1317—31no—
35C22×C3×D4—21no—
36C23×Dic3D411yescharacteristic
37C3×#123—41no—
38C3×#121—21no—
39C3×#121—21no—
40C3×#120—41no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—41no—
47C2×#217D411yescharacteristic
48#935—42no—
49#935—42no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—41no—
56C2×C4×Dic3—21no—
57C3×#107—41no—
58C3×#102—41no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62#834—42no—
63#832—41no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71#830—21no—
72#808—41no—
73#534C2×D311yescharacteristic
74C2×#123—61no—
75#481—61no—
76#480—21no—
77C2×#99—61no—
78#400—61no—
79#400—61no—
80#399—31no—
81#397C2×D311yescharacteristic
82#389—61no—
83#388—61no—
84#387—61no—
85#385—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×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125#217—22no—
126#217—42no—
127#217—44no—
128#217—41no—
129#217—41no—
130C4⋊Dic3—42no—
131C4⋊Dic3—41no—
132Dic3⋊C4—42no—
133Dic3⋊C4—44no—
134Dic3⋊C4—81no—
135Dic3⋊C4—81no—
136C4×Dic3—81no—
137C2×C3⋊C8—41no—
138C22×D4—21no—
139C23×C4—31no—
140#123—41no—
141#123—122no—
142#123—122no—
143#121—21no—
144#121—21no—
145#120—41no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—121no—
152C2×C4⋊C4#6711yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—121no—
158C2×C22⋊C4C22×D311yescharacteristic
159C2×C42—61no—
160#107—41no—
161#106—121no—
162#102—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#6711yescharacteristic
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—81no—
184C23×C3—81no—
185C23×C3#12011yescharacteristic
186C23×C3#13511yescharacteristic
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C3×D4—81no—
189C3×D4—81no—
190C3×D4—81no—
191C3×D4—81no—
192C3×D4—81no—
193C3×D4—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
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×Dic3—22no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C3×C8—81no—
225C3⋊C8—81no—
226C24—21no—
227C24D3×D411yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—121no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4C2×#6711yescharacteristic
250C2×C8—41no—
251C2×C8—121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—121no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—124no—
259C4⋊C4—241no—
260C4⋊C4—241no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—62no—
267C22⋊C4—122no—
268C22⋊C4—124no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C42—241no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3—82no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—161no—
286C22×C3#44911yescharacteristic
287C22×C3#45311yescharacteristic
288C22×C3B(4,𝔽2)11yescharacteristic
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4—81no—
293C3×C4—81no—
294Dic3—42no—
295Dic3—81no—
296Dic3—82no—
297Dic3—81no—
298Dic3—81no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—81no—
307C23—81no—
308C23#91111yescharacteristic
309C23#91211yescharacteristic
310C23#93711yescharacteristic
311D4—81no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—62no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4—241no—
347C8—81no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—161no—
354C2×C3#160011yescharacteristic
355C2×C3#219811yescharacteristic
356C2×C3#220111yescharacteristic
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—161no—
371C22#503211yescharacteristic
372C22#515811yescharacteristic
373C22#519811yescharacteristic
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—122no—
380C4—241no—
381C4—242no—
382C4—241no—
383C4—241no—
384C3#1218911yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2#7096911yescharacteristic
391C2#7099211yescharacteristic
392C2#7100211yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1607C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#534C22
hypercenter#1607D3
hypocenterC3#12189
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1607

Complements

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

NH ≅ G/Ntype
1. #77617C2semidirect
5. #67649C2semidirect
6. #67451C2semidirect
7. #67445C2semidirect
384. C3#12189semidirect

Maximal subgroups

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

subgroupindex
#776172
C3×#16072
#718492
#718442
#676492
#674512
#674452
#121893