Finite Group Databasebeta

← #10708201

Subgroups — #10708201

Order 768.

386 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 57 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#78008C211yesmaximal, characteristic
2#69713C211yesmaximal, characteristic, complemented
3#68978C211yesmaximal, characteristic, complemented
4#68138C211yesmaximal, characteristic
5C3×#1346C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic
7#67453C211yesmaximal, characteristic, complemented
8#12249—31nomaximal, Sylow
9C2×#898—21no—
10#5500C2211yescharacteristic
11C3×#398C2211yescharacteristic
12#5121—21no—
13#4994—21no—
14#4994—21no—
15#4991C2211yescharacteristic
16#4975C2211yescharacteristic
17#4971—21no—
18#4731—21no—
19#4676—21no—
20#4672—21no—
21#4657—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2823—31no—
29#2060—31no—
30#1611—31no—
31#1599—31no—
32#1346D311yescharacteristic, complemented
33#1345—31no—
34#1325—31no—
35C22×D12D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3D411yescharacteristic
56C2×#204—41no—
57#898—42no—
58#898—42no—
59#890—41no—
60#888—41no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95D411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#821—41no—
68#818—41no—
69#817—41no—
70#810—41no—
71#808—41no—
72#544—31no—
73C2×#126—61no—
74#483—61no—
75#450—61no—
76#428—61no—
77#398—61no—
78#398C2×D311yescharacteristic
79#396—61no—
80#396—61no—
81#396—61no—
82#394—31no—
83#394—61no—
84#384—31no—
85#343C2×D311yescharacteristic
86#339—61no—
87#328—61no—
88#328—61no—
89#324—31no—
90#324C2×D311yescharacteristic
91C23×D3—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×D12—41no—
101C2×D12—81no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C3×M16—22no—
105C3×M16—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112#217—41no—
113(C2×D3)⋊C4—42no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—81no—
118(C2×D3)⋊C4—81no—
119(C2×D3)⋊C4—81no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—42no—
125C4⋊Dic3—41no—
126Dic3⋊C4—42no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129Dic3⋊C4—81no—
130Dic3⋊C4—81no—
131Dic3⋊C4—81no—
132C4×Dic3—21no—
133C4×Dic3—41no—
134C4×Dic3SD1612yescomplemented
135C2×C3⋊C8—41no—
136#204—82no—
137C22×D4—31no—
138C2×SD16—121no—
139C2×M16—21no—
140C4⋊Q8—121no—
141#126—122no—
142#126—122no—
143#125—121no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#117—61no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4D1211yescharacteristic
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—31no—
163#107—121no—
164#106—121no—
165#106—121no—
166#103—121no—
167#102—121no—
168C22⋊C8—21no—
169C22⋊C8—61no—
170#95—21no—
171#95—31no—
172#95—61no—
173#95D1211yescharacteristic
174C23×C3#12011yescharacteristic
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4C2×SD1611yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207D12—42no—
208D12—81no—
209Dic6—81no—
210Dic6—81no—
211C3×C8—42no—
212C3×C8—81no—
213C3⋊C8—81no—
214C24—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×D4—121no—
218C2×D4—241no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227SD16—242no—
228M16—22no—
229M16—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—121no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—122no—
245C4⋊C4—122no—
246C4⋊C4—241no—
247C4⋊C4—241no—
248C4⋊C4—241no—
249C22⋊C4—122no—
250C22⋊C4—122no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—241no—
255C22⋊C4—241no—
256C22⋊C4—241no—
257C42—32no—
258C42—61no—
259C42—121no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3#45011yescharacteristic
263C22×C3#45611yescharacteristic
264C22×C3B(4,𝔽2)11yescharacteristic
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C2×D3—81no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—42no—
279Dic3—81no—
280Dic3—81no—
281Dic3—81no—
282Dic3—81no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23—121no—
289C23—121no—
290C23#88211yescharacteristic
291Q8—241no—
292Q8—241no—
293D4—122no—
294D4—241no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—62no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#91111yescharacteristic
319C2×C4D3×SD1611yescharacteristic
320C8—42no—
321C8—81no—
322C8—241no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3#160411yescharacteristic
326C2×C3#220011yescharacteristic
327C2×C3#220311yescharacteristic
328D3—161no—
329C22—21no—
330C22—21no—
331C22—241no—
332C22—241no—
333C22—241no—
334C22—241no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#474111yescharacteristic
339C22#477611yescharacteristic
340C22#482611yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—122no—
348C4—241no—
349C4—241no—
350C4—241no—
351C4—241no—
352C3#1224911yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—481no—
356C2#6829511yescharacteristic
357C2#6913911yescharacteristic
358C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1346D3
hypocenterC3#12249
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
2. #69713C2semidirect
3. #68978C2semidirect
5. C3×#1346C2semidirect
7. #67453C2semidirect
32. #1346D3semidirect
134. C4×Dic3SD16semidirect
352. C3#12249semidirect

Maximal subgroups

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

subgroupindex
#780082
#697132
#689782
#681382
C3×#13462
#674932
#674532
#122493