Finite Group Databasebeta

← #10708200

Subgroups — #10708200

Order 768.

386 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 59 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#74374C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic, complemented
3#68140C211yesmaximal, characteristic, complemented
4#68138C211yesmaximal, characteristic
5C3×#1325C211yesmaximal, characteristic, complemented
6#67458C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic
8#12249—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C2×#876—21no—
11C3×#384C2211yescharacteristic
12#4994—21no—
13#4994—21no—
14#4994—21no—
15#4991—21no—
16#4991C2211yescharacteristic
17#4981—21no—
18#4981C2211yescharacteristic
19#4964—21no—
20#4731—21no—
21#4721—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4530C2211yescharacteristic
25#4523—21no—
26#4467—21no—
27#4467—21no—
28#2823—31no—
29#2060—31no—
30#1611—31no—
31#1599—31no—
32#1346—31no—
33#1345—31no—
34#1325D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40C2×C3×C42D411yescharacteristic
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—
46C2×(C2×D3)⋊C4—21no—
47#926—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×#208—21no—
56C2×#204—41no—
57#890—41no—
58#888—41no—
59#876—42no—
60#876—42no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66C3⋊#95D411yescharacteristic
67#821—41no—
68#817—41no—
69#816—41no—
70#808—41no—
71#807—41no—
72#544—31no—
73C2×#126—61no—
74#483—61no—
75#450—61no—
76#428—61no—
77#398—31no—
78#398—61no—
79#396—61no—
80#396—61no—
81#396—61no—
82#394—31no—
83#394—61no—
84#384C2×D311yescharacteristic
85#343—31no—
86#339—61no—
87#328—61no—
88#328—61no—
89#324C2×D311yescharacteristic
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—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—81no—
109C3×C4⋊C4—81no—
110C3×C42—21no—
111C3×C42SD1612yescomplemented
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—41no—
125C4⋊Dic3—81no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—41no—
133#208—22no—
134#208—41no—
135C2×C3⋊C8—41no—
136#204—82no—
137C22×D4—31no—
138C2×SD16—121no—
139C2×M16—61no—
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—21no—
149C2×C4⋊C4—31no—
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×C42D1211yescharacteristic
163#107—121no—
164#106—121no—
165#106—121no—
166#103—121no—
167#102—121no—
168C22⋊C8—21no—
169C22⋊C8—21no—
170#95—31no—
171#95—31no—
172#95—61no—
173#95—61no—
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—22no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
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—41no—
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—81no—
212C3×C8—81no—
213C3⋊C8—42no—
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—62no—
229M16—121no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—121no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—121no—
242C4⋊C4—122no—
243C4⋊C4—122no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
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—21no—
258C42—121no—
259C42#20412yescomplemented
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—42no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278Dic3—81no—
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—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—121no—
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×C4C2×#20411yescharacteristic
319C2×C4#90811yescharacteristic
320C8—81no—
321C8—81no—
322C8—122no—
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#473111yescharacteristic
339C22#474111yescharacteristic
340C22#475211yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4—241no—
351C4—241no—
352C3#1224911yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—481no—
356C2#6813311yescharacteristic
357C2#6914711yescharacteristic
358C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1325D3
hypocenterC3#12249
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #69573C2semidirect
3. #68140C2semidirect
5. C3×#1325C2semidirect
6. #67458C2semidirect
34. #1325D3semidirect
111. C3×C42SD16semidirect
259. C42#204semidirect
352. C3#12249semidirect

Maximal subgroups

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

subgroupindex
#743742
#695732
#681402
#681382
C3×#13252
#674582
#674562
#122493