Finite Group Databasebeta

← #10693023

Subgroups — #10693023

Order 768.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#82501C211yesmaximal, characteristic
2#79286C211yesmaximal, characteristic
3C3×#1487C211yesmaximal, characteristic, complemented
4#71622C211yesmaximal, characteristic, complemented
5#70527C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9483—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5231C412yes—
14C2×#823—21no—
15#4993C412yes—
16#4989—22no—
17#4939—21no—
18#4936C2211yescharacteristic
19#4928—21no—
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4834—22no—
23D3×C22⋊C8—22no—
24C8×D12—22no—
25C3×Q8⋊C8—22no—
26#4554—21no—
27#4535—22no—
28#2972—31no—
29#2580—31no—
30#1800—31no—
31#1487—31no—
32#1487D311yescharacteristic, complemented
33#1286—31no—
34#1286—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12D411yescharacteristic
37C22×C4×D3D412yes—
38C2×C3×M16—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4C2×C412yes—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—22no—
53C2×#208—21no—
54C2×D3×C8—24no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8D412yes—
64C3⋊#95—21no—
65C3⋊#95—22no—
66C3⋊#95C2×C412yes—
67#830—21no—
68#823—42no—
69(C2×D3)⋊C8—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8D412yes—
74C2×C4×Q8C2×D311yescharacteristic
75C2×C4×D4—31no—
76C8×D4—62no—
77C8×D4—62no—
78#431—31no—
79#431C2×D311yescharacteristic
80#423—31no—
81#423C2×D311yescharacteristic
82C2×#100—61no—
83C2×C22⋊C8—62no—
84#399—32no—
85#393—32no—
86#390—61no—
87#389—61no—
88#385—62no—
89#382—31no—
90#343—61no—
91Q8⋊C8—22no—
92Q8⋊C8—62no—
93C23×D3C22⋊C412yes—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—42no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×Dic3—22no—
103C22×Dic3C22⋊C412yes—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12SD1612yes—
109C2×D12Q1612yes—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—82no—
113C3×M16—41no—
114C3×M16—41no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—44no—
130(C2×D3)⋊C4—82no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—82no—
136#208—41no—
137#208—41no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—41no—
140D3×C8—84no—
141C22×Q8C4×D311yescharacteristic
142C22×D4—31no—
143C23×C4—32no—
144C2×M16—21no—
145C2×M16—61no—
146C22×C8—64no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×D4—61no—
152C4×D4—61no—
153C4×D4—61no—
154C4×D4—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4C4×D311yescharacteristic
160C2×C22⋊C4—32no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#100—122no—
172C22⋊C8—21no—
173C22⋊C8—64no—
174C22⋊C8—61no—
175C4×C8—32no—
176C4×C8#6712yes—
177#95—32no—
178#95—62no—
179#95—61no—
180C23×C3C2×C22⋊C411yescharacteristic
181C22×D3—22no—
182C22×D3—22no—
183C22×D3—42no—
184C22×D3—42no—
185C22×D3#10314yes—
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—41no—
189C3×Q8—41no—
190C2×C3×C4—22no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4C2×Q1611yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4#13511yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213D12—22no—
214D12—22no—
215D12—22no—
216D12—22no—
217C4×D3—44no—
218C4×D3—82no—
219C3×C8—42no—
220C3×C8—81no—
221C3⋊C8—42no—
222C3⋊C8—81no—
223C24—32no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—42no—
227C2×D4—32no—
228C2×D4—32no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—32no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—242no—
240C22×C4C2×C4×D311yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243M16—41no—
244M16—41no—
245M16—121no—
246M16—121no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—62no—
250C2×C8—121no—
251C2×C8—244no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—242no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—124no—
264C22⋊C4—242no—
265C42—41no—
266C42—41no—
267C42#23511yescharacteristic
268C42D3×D411yescharacteristic
269C42#23711yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3—21no—
272C22×C3—21no—
273C22×C3#38611yescharacteristic
274C22×C3C2×#10311yescharacteristic
275C22×C3#41611yescharacteristic
276C2×D3—24no—
277C2×D3—42no—
278C2×D3—42no—
279C2×D3—44no—
280C2×D3—82no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—42no—
286C3×C4—81no—
287C3×C4#46511yescharacteristic
288C3×C4#46511yescharacteristic
289C3×C4#46711yescharacteristic
290C3×C4#46711yescharacteristic
291Dic3—42no—
292Dic3—82no—
293C23—34no—
294C23—62no—
295C23—62no—
296C23—122no—
297C23—122no—
298C23D3×C22⋊C411yescharacteristic
299Q8—22no—
300Q8—22no—
301Q8—41no—
302Q8—41no—
303D4—62no—
304D4—62no—
305D4—62no—
306D4—62no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—62no—
316C2×C4—124no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—242no—
322C2×C4#67⋊C411yescharacteristic
323C2×C4#88311yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4D3×SD1611yescharacteristic
326C2×C4#91611yescharacteristic
327C2×C4D3×Q1611yescharacteristic
328C2×C4#91911yescharacteristic
329C2×C4C4×#6711yescharacteristic
330C2×C4#93711yescharacteristic
331C2×C4#93811yescharacteristic
332C8—42no—
333C8—81no—
334C8—122no—
335C8—241no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3#189111yescharacteristic
339C2×C3#189211yescharacteristic
340C2×C3#191111yescharacteristic
341D3—44no—
342C22—21no—
343C22—21no—
344C22—64no—
345C22—124no—
346C22—122no—
347C22—122no—
348C22—242no—
349C22D3×#10311yescharacteristic
350C22#480411yescharacteristic
351C22#495611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—81no—
358C4—122no—
359C4—242no—
360C4#480711yescharacteristic
361C4#481311yescharacteristic
362C4#517111yescharacteristic
363C4#518811yescharacteristic
364C3#948311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—124no—
368C2#6999211yescharacteristic
369C2#7027011yescharacteristic
370C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1487D3
hypocenterC3#9483
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. C3×#1487C2semidirect
4. #71622C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
32. #1487D3semidirect
364. C3#9483semidirect

Maximal subgroups

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

subgroupindex
#825012
#792862
C3×#14872
#716222
#705272
#677382
#677132
#94833