Finite Group Databasebeta

← #10690236

Subgroups — #10690236

Order 768.

768 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2315C211yesmaximal, characteristic
2C2×#4953C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic
4#69501C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9025—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×#217—21no—
10C22×C4⋊Dic3C2211yescharacteristic
11C3×#385—24no—
12C3×#380—21no—
13C3×#380C2212yes—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#4953—22no—
18#4953C414yescomplemented
19#4529—24no—
20#4525—22no—
21C2×#397—31no—
22#2315D311yescharacteristic
23#1823—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C23×Dic3—21no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4D414yes—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—42no—
39C2×C3×C22⋊C4—42no—
40C2×C3×C22⋊C4C2×C412yes—
41C2×C3×C22⋊C4D412yes—
42C2×#217—24no—
43C2×#217—42no—
44C2×#217—42no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C412yescomplemented
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3⋊#95—24no—
53C3⋊#95—21no—
54C3⋊#95C2×C412yescomplemented
55#830—22no—
56#830—22no—
57C22×C4⋊C4—31no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—61no—
60C22×C22⋊C4C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#397—34no—
63#397—62no—
64#385—24no—
65#380—21no—
66#380C2×D312yes—
67C2×#95—31no—
68C2×#95—61no—
69#340—64no—
70#323—62no—
71C24×C3—21no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—42no—
76C24×C3—42no—
77C24×C3C22⋊C412yes—
78C24×C3D812yes—
79C24×C3SD1612yes—
80C24×C3C2×D411yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—24no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4—44no—
87C22×C3×C4—82no—
88C22×C3×C4—82no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4#4112yes—
94C22×Dic3—22no—
95C22×Dic3—22no—
96C22×Dic3—42no—
97C22×Dic3—42no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C3×C4⋊C4—44no—
107C3×C4⋊C4—82no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4—24no—
110C3×C22⋊C4—44no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113#217—48no—
114C4⋊Dic3—24no—
115C4⋊Dic3—42no—
116C2×C3⋊C8—44no—
117C2×C3⋊C8—42no—
118C25#6711yescharacteristic
119C23×C4—22no—
120C23×C4—61no—
121C23×C4—61no—
122C23×C4#6711yescharacteristic
123C23×C4C22×D311yescharacteristic
124C22×C8—62no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4#6714yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—64no—
138C2×C22⋊C4—122no—
139C2×C22⋊C4—122no—
140C2×C22⋊C4C4×D312yes—
141C2×C22⋊C4D1212yes—
142C22⋊C8—62no—
143C22⋊C8—62no—
144#95—22no—
145#95—22no—
146#95—32no—
147#95—64no—
148#95—61no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—44no—
160C23×C3—44no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—82no—
164C23×C3—82no—
165C23×C3—81no—
166C23×C3—81no—
167C23×C3#10214yes—
168C23×C3C2×C22⋊C411yescharacteristic
169C23×C3#12011yescharacteristic
170C23×C3#12311yescharacteristic
171C23×C3C2×D811yescharacteristic
172C23×C3C2×SD1611yescharacteristic
173C23×C3AGL(1,ℤ/8)11yescharacteristic
174C23×C3#13711yescharacteristic
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—44no—
185C2×C3×C4—81no—
186C2×C3×C4—88no—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—82no—
196C2×Dic3—82no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C3⋊C8—82no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—42no—
207C24—42no—
208C24(C2×D3)⋊C412yes—
209C24#21312yes—
210C24#21512yes—
211C24C2×#6711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—122no—
223C22×C4—122no—
224C22×C4—121no—
225C22×C4—121no—
226C22×C4—121no—
227C22×C4—121no—
228C22×C4—121no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4(C2×D3)⋊C412yes—
233C22×C4C2×C4×D311yescharacteristic
234C22×C4C2×D1211yescharacteristic
235C22×C4#23512yes—
236C22×C4C2×#6712yes—
237C2×C8—124no—
238C2×C8—122no—
239C4⋊C4—44no—
240C4⋊C4—64no—
241C4⋊C4—82no—
242C4⋊C4—122no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—128no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—24no—
258C22×C3—44no—
259C22×C3—44no—
260C22×C3—44no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—82no—
264C22×C3—82no—
265C22×C3—82no—
266C22×C3#38611yescharacteristic
267C22×C3C2×#10211yescharacteristic
268C22×C3#41711yescharacteristic
269C22×C3#44711yescharacteristic
270C22×C3#44811yescharacteristic
271C22×C3#45512yes—
272C22×C3#45812yes—
273C22×C3#48011yescharacteristic
274C22×C3#48111yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—84no—
279Dic3—82no—
280Dic3—82no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—44no—
292C23—44no—
293C23—42no—
294C23—42no—
295C23—82no—
296C23—82no—
297C23—81no—
298C23—81no—
299C23#80914yes—
300C23C2×(C2×D3)⋊C411yescharacteristic
301C23#92911yescharacteristic
302C23C2×#21311yescharacteristic
303C23#93211yescharacteristic
304C23C2×#21511yescharacteristic
305C23#94211yescharacteristic
306C23#95311yescharacteristic
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—81no—
318C2×C4—88no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—242no—
326C2×C4—242no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4C4×#6712yes—
332C2×C4#93012yes—
333C8—242no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—44no—
339C2×C3#160712yes—
340C2×C3#160812yes—
341C2×C3#187411yescharacteristic
342C2×C3#188511yescharacteristic
343C2×C3#189011yescharacteristic
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—24no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22C2×#80911yescharacteristic
362C22#496611yescharacteristic
363C22#496711yescharacteristic
364C22#496811yescharacteristic
365C22#521311yescharacteristic
366C22#521711yescharacteristic
367C22#522711yescharacteristic
368C22#524212yes—
369C22#524312yes—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—84no—
374C4—242no—
375C4—242no—
376C3#902511yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#7156011yescharacteristic
383C2#7164911yescharacteristic
384C2#7165511yescharacteristic
385C2#7184312yes—
386C2#7189012yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2315C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2315D3
hypocenterC3#9025
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
4. #69501C2semidirect
5. #67624C2semidirect
18. #4953C4semidirect
48. C2×C4⋊Dic3C2×C4semidirect
54. C3⋊#95C2×C4semidirect
376. C3#9025semidirect

Maximal subgroups

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

subgroupindex
C3×#23152
C2×#49532
C2×#45252
#695012
#676242
#90253