Finite Group Databasebeta

← #10693025

Subgroups — #10693025

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#79286C211yesmaximal, characteristic
2#74190C211yesmaximal, characteristic
3#70770C211yesmaximal, characteristic, complemented
4#70527C211yesmaximal, characteristic
5#70500C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9483—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#823—21no—
15#4988—21no—
16#4984—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4834—22no—
20D3×C22⋊C8—22no—
21C8×D12—22no—
22#4677C412yes—
23#4673C412yes—
24#4668—22no—
25#4554—21no—
26#4484—22no—
27#4456—22no—
28#2972—31no—
29#2580—31no—
30#1800—31no—
31#1487—31no—
32#1487—31no—
33#1286—31no—
34#1286D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C22×C4×D3D412yes—
37C22×Dic6C2×C411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4C2×C412yes—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C2×D3×C8—24no—
51C4×D12—21no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8D412yes—
64C3×#95—21no—
65C3×#95C2×C412yes—
66C3⋊#95—22no—
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×Q8—31no—
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#382C2×D311yescharacteristic
90#343—61no—
91Q8⋊C8—62no—
92Q8⋊C8—62no—
93C23×D3C22⋊C412yes—
94C22×C3×C4—22no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3C22⋊C412yes—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12SD1612yes—
106C2×D12Q1612yes—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—82no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6—42no—
113C3×M16—41no—
114C3×M16—41no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—44no—
126(C2×D3)⋊C4—82no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—82no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—41no—
135C4×Dic3—41no—
136#208—41no—
137#208—41no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—41no—
140D3×C8—84no—
141C22×Q8—31no—
142C22×D4—31no—
143C23×C4—32no—
144C2×M16—21no—
145C2×M16—61no—
146C22×C8—64no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151C4×D4—61no—
152C4×D4—61no—
153C4×D4—61no—
154C4×D4—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—62no—
160C2×C22⋊C4—32no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C42—61no—
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×C8D1212yes—
177#95—21no—
178#95—62no—
179#95C4×D312yes—
180C23×C3C2×C22⋊C411yescharacteristic
181C22×D3—22no—
182C22×D3—22no—
183C22×D3—42no—
184C22×D3—42no—
185C22×D3#10314yes—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4C2×SD1611yescharacteristic
198C2×C3×C4C2×Q1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209D12—22no—
210D12—22no—
211D12—22no—
212D12—22no—
213C4×D3—44no—
214C4×D3—82no—
215Dic6—22no—
216Dic6—22no—
217Dic6—41no—
218Dic6—41no—
219C3×C8—42no—
220C3×C8—81no—
221C3⋊C8—42no—
222C3⋊C8—81no—
223C24—32no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—122no—
227C2×D4—32no—
228C2×D4—32no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C22×C4—22no—
234C22×C4—32no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—61no—
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—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—61no—
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—121no—
266C42—121no—
267C42C2×D1211yescharacteristic
268C42#23511yescharacteristic
269C42D3×D411yescharacteristic
270C42#23911yescharacteristic
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—82no—
286C3×C4#46511yescharacteristic
287C3×C4#46511yescharacteristic
288C3×C4#46711yescharacteristic
289C3×C4#46711yescharacteristic
290Dic3—42no—
291Dic3—42no—
292Dic3—81no—
293C23—34no—
294C23—62no—
295C23—62no—
296C23—122no—
297C23—122no—
298C23D3×C22⋊C411yescharacteristic
299Q8—62no—
300Q8—62no—
301Q8—121no—
302Q8—121no—
303D4—62no—
304D4—62no—
305D4—62no—
306D4—62no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—124no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—242no—
322C2×C4C4×D1211yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#89311yescharacteristic
326C2×C4#89411yescharacteristic
327C2×C4#89511yescharacteristic
328C2×C4#91211yescharacteristic
329C2×C4D3×SD1611yescharacteristic
330C2×C4#91611yescharacteristic
331C2×C4D3×Q1611yescharacteristic
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—
349C22#466511yescharacteristic
350C22#477211yescharacteristic
351C22D3×#10311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—122no—
358C4—122no—
359C4—241no—
360C4#477811yescharacteristic
361C4#481311yescharacteristic
362C4#486411yescharacteristic
363C4#488711yescharacteristic
364C3#948311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—124no—
368C2#6870511yescharacteristic
369C2#6888211yescharacteristic
370C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9483
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #70770C2semidirect
5. #70500C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
34. #1286D3semidirect
364. C3#9483semidirect

Maximal subgroups

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

subgroupindex
#792862
#741902
#707702
#705272
#705002
C3×#12862
#676312
#94833