Finite Group Databasebeta

← #10695890

Subgroups — #10695890

Order 768.

562 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#4968C211yesmaximal, characteristic, complemented
2C2×#4967C211yesmaximal, characteristic, complemented
3#78641C211yesmaximal, characteristic
4C3×#1821C211yesmaximal, characteristic, complemented
5#69970C211yesmaximal, characteristic, complemented
6#69970C211yesmaximal, characteristic, complemented
7#67688C211yesmaximal, characteristic, complemented
8#9992—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C2×#830C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5135—22no—
17#4968—22no—
18#4968—22no—
19#4967—22no—
20#4967—22no—
21C2×#809—21no—
22C2×#809—21no—
23C2×#808—21no—
24C2×#807—21no—
25#4735—21no—
26#4735—21no—
27#4732—21no—
28#4732—21no—
29C3×#343—22no—
30#4468—22no—
31#4468—22no—
32#3192—31no—
33C2×#481—31no—
34C2×#480—31no—
35#1821D311yescharacteristic, complemented
36#1471—31no—
37#1471—31no—
38#1298—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×M16D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C2×D24—22no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72#830—21no—
73#830—21no—
74#830D412yes—
75#830D412yes—
76#821—22no—
77#821—22no—
78#817—22no—
79#809—24no—
80#809—24no—
81#808—24no—
82#807—24no—
83C2×#121—31no—
84C2×#121—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86#481—61no—
87#481—61no—
88#481—62no—
89#481—62no—
90#480—62no—
91#480—62no—
92#469—62no—
93#447—61no—
94#447—61no—
95C2×#107—61no—
96C2×#106—61no—
97C2×#102—61no—
98C2×#102—61no—
99#407C2×D311yescharacteristic
100#407C2×D311yescharacteristic
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#343—22no—
104#323—62no—
105#323—62no—
106C24×C3#4111yescharacteristic
107C23×D3—21no—
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—82no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—82no—
125C2×#67—82no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×D12—42no—
134C3×M16—42no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—42no—
153C2×C3⋊C8—42no—
154D24—82no—
155C22×D4—31no—
156C22×D4—31no—
157C22×D4—61no—
158C22×D4—61no—
159C23×C4—21no—
160C23×C4C22×D311yescharacteristic
161C2×D8—62no—
162C2×M16—21no—
163C2×M16#6712yes—
164C22×C8—61no—
165C22×C8—61no—
166#121—61no—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173#121—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4D1212yes—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184#107—64no—
185#106—64no—
186#106—62no—
187#102—64no—
188#102—64no—
189#102—62no—
190#102—62no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—32no—
196C22⋊C8—32no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3#12111yescharacteristic
203C23×C3#12111yescharacteristic
204C23×C3#12111yescharacteristic
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—82no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12711yescharacteristic
227#67—82no—
228#67—82no—
229#67—82no—
230#67—82no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235D12—42no—
236D12—42no—
237D12—42no—
238D12—42no—
239C3×C8—42no—
240C3×C8—81no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C24—61no—
244C24—61no—
245C24#23511yescharacteristic
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C2×D4—122no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—242no—
257C2×D4—242no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—82no—
265C22×C4C2×D1211yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4C2×#6711yescharacteristic
271D8—242no—
272M16—42no—
273M16—42no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—122no—
279C2×C8—122no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—44no—
300C22×C3#39211yescharacteristic
301C22×C3#47112yes—
302C22×C3#47212yes—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—162no—
310C2×D3—162no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4#45812yes—
316Dic3—81no—
317Dic3—81no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—121no—
322C23—121no—
323C23—122no—
324C23—122no—
325C23—241no—
326C23—241no—
327C23—241no—
328C23—241no—
329C23#88411yescharacteristic
330C23#88411yescharacteristic
331C23#93011yescharacteristic
332D4—122no—
333D4—122no—
334D4—122no—
335D4—122no—
336D4—242no—
337D4—242no—
338D4—242no—
339D4—242no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4#88211yescharacteristic
355C2×C4#88211yescharacteristic
356C2×C4#93711yescharacteristic
357C2×C4#94011yescharacteristic
358C8—42no—
359C8—81no—
360C8—122no—
361C8—122no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—41no—
365C2×C3#201811yescharacteristic
366C2×C3#201911yescharacteristic
367C2×C3#202211yescharacteristic
368D3—161no—
369D3—161no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—44no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—482no—
385C22—482no—
386C22#474912yes—
387C22#475412yes—
388C22#496011yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—241no—
394C4—241no—
395C4#505312yes—
396C3#999211yescharacteristic, complemented, Sylow
397C2—22no—
398C2—22no—
399C2—41no—
400C2—481no—
401C2—481no—
402C2#6953911yescharacteristic
403C2#7009011yescharacteristic
404C2#7011711yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1821D3
hypocenterC3#9992
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
1. C2×#4968C2semidirect
2. C2×#4967C2semidirect
4. C3×#1821C2semidirect
5. #69970C2semidirect
6. #69970C2semidirect
7. #67688C2semidirect
35. #1821D3semidirect
396. C3#9992semidirect

Maximal subgroups

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

subgroupindex
C2×#49682
C2×#49672
#786412
C3×#18212
#699702
#676882
#99923