Finite Group Databasebeta

← #10693071

Subgroups — #10693071

Order 768.

600 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2873C211yesmaximal, characteristic, complemented
2#82284C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic, complemented
4#69855C212yesmaximal, complemented
5#67666C212yesmaximal, complemented
6#9492—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×C4×D12C2211yescharacteristic
10C3×#405—24no—
11C3×C4×M16—21no—
12C3×C4×M16C2212yes—
13#5213—21no—
14#4991C411yescharacteristic
15#4989—21no—
16#4986C411yescharacteristic
17#4960C411yescharacteristic
18#4956—21no—
19#4949C411yescharacteristic
20#4940—21no—
21#4937C2211yescharacteristic
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4459—24no—
26#2873D311yescharacteristic, complemented
27#2598—31no—
28#1490—32no—
29#1464—31no—
30#1288—32no—
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C22×C4×D3—21no—
35C2×C3×M16—24no—
36C2×C3×M16—24no—
37C2×C3×M16—22no—
38C2×C3×M16—22no—
39C22×C3×C8—22no—
40C3×#117—22no—
41C3×#117—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#217—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—21no—
55C4×D12—22no—
56C4×D12—22no—
57C3×#97—22no—
58C3×#97—22no—
59C3×C4×C8—22no—
60C3×C4×C8D414yescomplemented
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95C2×C411yescharacteristic
66C3⋊#95C2×C411yescharacteristic
67#830—22no—
68C4⋊(C3⋊C8)—22no—
69#803—22no—
70C22×M16—21no—
71C2×C4×D4—31no—
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—24no—
75C4×M16—21no—
76C4×M16C2×D312yes—
77#400—31no—
78#399—31no—
79#394—31no—
80#392—31no—
81#388—61no—
82#386—61no—
83#386—61no—
84#385—61no—
85#383—31no—
86D4⋊C8—64no—
87#324—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C23×D3—21no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C2×#67—42no—
102C2×#67—42no—
103C2×#67—81no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C2×D12—22no—
108C2×D12—42no—
109C2×C4×D3—42no—
110C2×C4×D3—81no—
111C2×C4×D3—81no—
112C3×M16—28no—
113C3×M16—24no—
114C3×M16—24no—
115C2×C3×C8—28no—
116C2×C3×C8—24no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42C2×D412yes—
124C3×C42#4112yes—
125#217—81no—
126(C2×D3)⋊C4—42no—
127(C2×D3)⋊C4—42no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—81no—
131C4⋊Dic3—22no—
132Dic3⋊C4—81no—
133C2×C3⋊C8—42no—
134C22×D4—31no—
135C22×D4—61no—
136C23×C4—61no—
137C23×C4#6711yescharacteristic
138C2×M16—22no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—22no—
142C22×C8—22no—
143C4×D4—62no—
144C4×D4—62no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4C4×D311yescharacteristic
150C2×C4⋊C4D1211yescharacteristic
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4C4×D311yescharacteristic
157C2×C22⋊C4D1211yescharacteristic
158C2×C42#6711yescharacteristic
159C2×C42C22×D311yescharacteristic
160C4⋊C8—62no—
161C22⋊C8—62no—
162#97—22no—
163#97—22no—
164#97—62no—
165C4×C8—22no—
166C4×C8#6714yescomplemented
167#95—31no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C22×D3—22no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4#10412yes—
200C2×C3×C4C4×D412yes—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12112yes—
203C2×C3×C4#12311yescharacteristic
204#67—84no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214D12—24no—
215D12—42no—
216C4×D3—84no—
217C3×C8—28no—
218C3⋊C8—82no—
219C24—61no—
220C24(C2×D3)⋊C411yescharacteristic
221C2×D4—62no—
222C2×D4—122no—
223C2×D4—122no—
224C2×D4—122no—
225C2×D4—241no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—122no—
234C22×C4—241no—
235C22×C4—241no—
236C22×C4(C2×D3)⋊C411yescharacteristic
237C22×C4(C2×D3)⋊C411yescharacteristic
238C22×C4(C2×D3)⋊C411yescharacteristic
239C22×C4C2×C4×D311yescharacteristic
240C22×C4C2×D1211yescharacteristic
241C22×C4C2×#6711yescharacteristic
242M16—28no—
243M16—24no—
244M16—24no—
245C2×C8—28no—
246C2×C8—24no—
247C2×C8—122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—62no—
251C4⋊C4—241no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—241no—
259C22⋊C4—241no—
260C42—21no—
261C42—21no—
262C42#23512yes—
263C42C2×#6712yes—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—42no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—81no—
271C22×C3#38611yescharacteristic
272C22×C3C2×#10411yescharacteristic
273C22×C3C2×#10411yescharacteristic
274C2×D3—42no—
275C2×D3—42no—
276C2×D3—42no—
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—22no—
282C3×C4—21no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4—41no—
288Dic3—81no—
289Dic3—81no—
290Dic3—81no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—62no—
296C23—121no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23—121no—
301C23—121no—
302C23#80512yes—
303C23C2×(C2×D3)⋊C411yescharacteristic
304D4—64no—
305D4—122no—
306D4—244no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—244no—
330C2×C4#80512yes—
331C2×C4C4×#6712yes—
332C2×C4#92911yescharacteristic
333C2×C4#93012yes—
334C2×C4#95311yescharacteristic
335C8—28no—
336C8—242no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—42no—
340C2×C3#189111yescharacteristic
341C2×C3#189611yescharacteristic
342C2×C3#191011yescharacteristic
343D3—82no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—81no—
351C22—122no—
352C22—122no—
353C22—122no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22—241no—
358C22C2×#80511yescharacteristic
359C22C2×#80511yescharacteristic
360C22#521311yescharacteristic
361C4—22no—
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—241no—
369C4—241no—
370C4—241no—
371C3#949211yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2—42no—
375C2—242no—
376C2#7153211yescharacteristic
377C2#7202511yescharacteristic
378C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2873C2
socleC22×C3#386
prime radical#4960C4
cosocleC2×C3×C42C23
hypercenter#2873D3
hypocenterC3#9492
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
1. C3×#2873C2semidirect
3. #71572C2semidirect
4. #69855C2semidirect
5. #67666C2semidirect
26. #2873D3semidirect
60. C3×C4×C8D4semidirect
166. C4×C8#67semidirect
371. C3#9492semidirect

Maximal subgroups

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

subgroupindex
C3×#28732
#822842
#715722
#698552
#676662
#94923