Finite Group Databasebeta

← #10706109

Subgroups — #10706109

Order 768.

438 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 71 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#78358C211yesmaximal, characteristic
2#68990C211yesmaximal, characteristic, complemented
3D3×#324C211yesmaximal, characteristic, complemented
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic, complemented
8#11852—31nomaximal, Sylow
9C2×#895C411yescharacteristic
10C2×D3×C4⋊C4C2211yescharacteristic
11C3×#400C2211yescharacteristic
12#5126—21no—
13#4991C2211yescharacteristic
14#4985C411yescharacteristic
15D3×C22⋊C8—21no—
16#4672C411yescharacteristic
17#4670—21no—
18#4669—21no—
19#4666C411yescharacteristic
20#4665—21no—
21#4665—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2591—31no—
29#1821—31no—
30C2×#324—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1345—31no—
34#1345—31no—
35C22×D12—21no—
36C22×D12C2×C411yescharacteristic
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4C2×C411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3—21no—
53C2×#208—21no—
54C2×D3×C8—41no—
55#895—41no—
56#895—41no—
57D3×C4⋊C4—41no—
58D3×C4⋊C4—41no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66#830—21no—
67(C2×D3)⋊C8—41no—
68(C2×D3)⋊C8—41no—
69(C2×D3)⋊C8—41no—
70C2×#121—31no—
71C22×C4⋊C4—31no—
72#407—61no—
73C2×C22⋊C8—61no—
74#400—31no—
75#400C2×D311yescharacteristic
76#394—31no—
77#394—31no—
78#388—61no—
79#388—61no—
80#386—61no—
81#386—61no—
82C4×C22⋊C4—31no—
83#343—31no—
84#343—61no—
85#343C2×D311yescharacteristic
86#324—31no—
87#324—61no—
88#324C2×D311yescharacteristic
89C23×D3—21no—
90C23×D3C22⋊C411yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3C22⋊C411yescharacteristic
98C22×Dic3C22⋊C411yescharacteristic
99C22×Dic3C22⋊C411yescharacteristic
100C2×D12—41no—
101C2×D12—42no—
102C2×D12—42no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—81no—
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—41no—
115C2×C4×D3—81no—
116C2×C4×D3—81no—
117C3×M16—41no—
118C3×M16—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—42no—
125(C2×D3)⋊C4—42no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—81no—
129(C2×D3)⋊C4—81no—
130C4⋊Dic3—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—81no—
134#208—41no—
135#208—41no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138D3×C8—82no—
139C22×D4—31no—
140C22×D4—61no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C2×M16—21no—
145C2×M16—61no—
146C22×C8—121no—
147#121—121no—
148#121—121no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—121no—
153C2×C4⋊C4—121no—
154C2×C4⋊C4C4×D311yescharacteristic
155C2×C4⋊C4C22×D311yescharacteristic
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C42—61no—
163C22⋊C8—21no—
164C22⋊C8—61no—
165C22⋊C8—121no—
166C22⋊C8—121no—
167C22⋊C8—121no—
168#95—21no—
169#95—21no—
170#95—31no—
171#95—61no—
172#95—61no—
173#95C4×D311yescharacteristic
174C23×C3C2×C22⋊C411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—22no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3#9911yescharacteristic
187C22×D3#9911yescharacteristic
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
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×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4#13511yescharacteristic
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212D12—82no—
213D12—82no—
214D12—82no—
215C4×D3—41no—
216C4×D3—41no—
217C4×D3—41no—
218C4×D3—41no—
219C4×D3—42no—
220C4×D3—81no—
221C4×D3—81no—
222C4×D3—81no—
223C3×C8—81no—
224C3×C8—81no—
225C3⋊C8—81no—
226C3⋊C8—81no—
227C24—31no—
228C24—61no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—122no—
233C2×D4—122no—
234C2×D4—241no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—121no—
250C22×C4—241no—
251C22×C4—241no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—41no—
256M16—41no—
257M16—121no—
258M16—121no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—242no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—122no—
270C22⋊C4—122no—
271C22⋊C4—122no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—241no—
275C22⋊C4—241no—
276C42—241no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3C2×#9911yescharacteristic
280C22×C3#41611yescharacteristic
281C22×C3#42111yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3#35512yes—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—81no—
302C3×C4—81no—
303Dic3—41no—
304Dic3—41no—
305Dic3—41no—
306Dic3—81no—
307C23—31no—
308C23—31no—
309C23—62no—
310C23—61no—
311C23—61no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23D3×C22⋊C411yescharacteristic
321D4—242no—
322D4—242no—
323D4—242no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
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—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—241no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4#91611yescharacteristic
355C2×C4#91911yescharacteristic
356C8—81no—
357C8—81no—
358C8—241no—
359C8—241no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#151611yescharacteristic
363C2×C3#212111yescharacteristic
364C2×C3C2×#35511yescharacteristic
365D3—22no—
366D3—41no—
367D3—161no—
368C22—21no—
369C22—21no—
370C22—32no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—122no—
377C22—122no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22D3×#9911yescharacteristic
386C22#480411yescharacteristic
387C22#482111yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4—121no—
395C4—121no—
396C4—121no—
397C4—241no—
398C3#1185211yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—62no—
402C2—121no—
403C2—481no—
404C2#6833811yescharacteristic
405C2#6914311yescharacteristic
406C2D3×#35511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC2×#895C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11852
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
2. #68990C2semidirect
3. D3×#324C2semidirect
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67493C2semidirect
7. #67456C2semidirect
32. #1353D3semidirect
398. C3#11852semidirect

Maximal subgroups

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

subgroupindex
#783582
#689902
D3×#3242
C3×#13532
#676522
#674932
#674562
#118523