Finite Group Databasebeta

← #10690599

Subgroups — #10690599

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79286C211yesmaximal, characteristic
2C2×#4710C211yesmaximal, characteristic
3#70786C211yesmaximal, characteristic, complemented
4#70767C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic
6C3×#1275C211yesmaximal, characteristic, complemented
7#67661C211yesmaximal, characteristic, complemented
8#9087—31nomaximal, Sylow
9C22×#204C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×#821C411yescharacteristic
16C2×#818C411yescharacteristic
17C2×#816C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4834—22no—
21D3×C22⋊C8—22no—
22#4710—22no—
23#4710—22no—
24C8×D12—22no—
25#4483—22no—
26#4469—22no—
27#4456—22no—
28#2972—31no—
29C2×#440—31no—
30#1479—31no—
31#1478—31no—
32#1461—31no—
33#1281—31no—
34#1275D311yescharacteristic, complemented
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6C2×C411yescharacteristic
38C22×C3×C8C2×C411yescharacteristic
39C22×C3×C8C2×C411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3—21no—
46C2×#204—22no—
47C2×#204—22no—
48C2×#204C814yescomplemented
49C2×D3×C8—24no—
50C4×D12—21no—
51C4×D12—21no—
52C4×D12D412yes—
53C4×D12D412yes—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#97—21no—
63C3×#97—21no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#830—21no—
67#830D412yes—
68#821—22no—
69(C2×D3)⋊C8—24no—
70#818—22no—
71#816—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C22×SD16—31no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79#440—62no—
80#440—62no—
81C8×D4—62no—
82C8×D4—62no—
83#431—31no—
84#431—31no—
85#431C2×D311yescharacteristic
86#431C2×D311yescharacteristic
87C2×#107—31no—
88C2×#103—31no—
89C2×#102—31no—
90C2×C22⋊C8—62no—
91C2×#97C2×D311yescharacteristic
92Q8⋊C8—62no—
93D4⋊C8—62no—
94#324—62no—
95C23×D3—21no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12C22⋊C412yes—
106C2×D12C2×C812yes—
107C2×D12M1612yes—
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—82no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6C2×C812yescomplemented
117C2×C3×C8—21no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—22no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—41no—
124C2×C3×C8C2×C812yescomplemented
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138C4×Dic3—41no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141#204—28no—
142D3×C8—84no—
143C22×Q8—31no—
144C22×D4—31no—
145C23×C4—61no—
146C2×SD16—34no—
147C2×SD16—62no—
148C2×SD16—62no—
149C22×C8—64no—
150C22×C8C4×D311yescharacteristic
151C22×C8C4×D311yescharacteristic
152C4×Q8—61no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—32no—
157C4×D4—32no—
158C4×D4—61no—
159C4×D4—61no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#103—62no—
172#102—62no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—32no—
176C22⋊C8—64no—
177C22⋊C8—61no—
178#97—21no—
179#97—21no—
180C4×C8—21no—
181C4×C8—61no—
182C4×C8—61no—
183C4×C8D1212yes—
184C23×C3C4×D411yescharacteristic
185C22×D3—22no—
186C22×D3—22no—
187C22×D3—22no—
188C22×D3—81no—
189C22×D3—81no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12311yescharacteristic
198C2×C3×C4C22×C811yescharacteristic
199C2×C3×C4C2×M1611yescharacteristic
200C2×C3×C4#13111yescharacteristic
201C2×C3×C4#13111yescharacteristic
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209D12—22no—
210D12—22no—
211D12—22no—
212D12C22⋊C814yes—
213C4×D3—44no—
214C4×D3—82no—
215Dic6—22no—
216Dic6—22no—
217Dic6—41no—
218Dic6—41no—
219C3×C8—22no—
220C3×C8—42no—
221C3×C8—41no—
222C3×C8—81no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225C24—61no—
226C2×Q8—32no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—32no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—61no—
243C22×C4—242no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4#23911yescharacteristic
247SD16—68no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—62no—
256C2×C8—121no—
257C2×C8—244no—
258C2×C8D3×C812yescomplemented
259C4⋊C4—32no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C42—121no—
268C42—121no—
269C42C2×D1211yescharacteristic
270C42#23511yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#38611yescharacteristic
276C22×C3C8×D411yescharacteristic
277C22×C3#43511yescharacteristic
278C2×D3—24no—
279C2×D3—42no—
280C2×D3—42no—
281C2×D3—82no—
282C2×D3—82no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4C2×C22⋊C811yescharacteristic
288C3×C4#40811yescharacteristic
289C3×C4C8×D411yescharacteristic
290C3×C4#43511yescharacteristic
291Dic3—42no—
292Dic3—42no—
293Dic3—81no—
294C23—62no—
295C23—62no—
296C23—62no—
297C23—241no—
298C23—241no—
299C23#89311yescharacteristic
300Q8—62no—
301Q8—62no—
302Q8—121no—
303Q8—121no—
304D4—34no—
305D4—62no—
306D4—62no—
307D4—62no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—124no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4C4×D1211yescharacteristic
320C2×C4D3×C22⋊C411yescharacteristic
321C2×C4#88211yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#89411yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4C2×D3×C811yescharacteristic
326C2×C4#90111yescharacteristic
327C2×C4D3×M1611yescharacteristic
328C2×C4#90711yescharacteristic
329C8—22no—
330C8—42no—
331C8—41no—
332C8—81no—
333C8—122no—
334C8—241no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3#186911yescharacteristic
338C2×C3#190211yescharacteristic
339C2×C3#190411yescharacteristic
340D3—44no—
341C22—21no—
342C22—21no—
343C22—64no—
344C22—122no—
345C22—122no—
346C22—242no—
347C22—242no—
348C22#466511yescharacteristic
349C22#483411yescharacteristic
350C22#483611yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—122no—
356C4—122no—
357C4—241no—
358C4C8×D1211yescharacteristic
359C4#470611yescharacteristic
360C4D3×C22⋊C811yescharacteristic
361C4#472711yescharacteristic
362C3#908711yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—124no—
366C2#6866811yescharacteristic
367C2#6870411yescharacteristic
368C2#6872511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1275C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1275D3
hypocenterC3#9087
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
3. #70786C2semidirect
4. #70767C2semidirect
6. C3×#1275C2semidirect
7. #67661C2semidirect
34. #1275D3semidirect
48. C2×#204C8semidirect
116. C2×Dic6C2×C8semidirect
124. C2×C3×C8C2×C8semidirect
258. C2×C8D3×C8semidirect
362. C3#9087semidirect

Maximal subgroups

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

subgroupindex
#792862
C2×#47102
#707862
#707672
#707562
C3×#12752
#676612
#90873