Finite Group Databasebeta

← #10698540

Subgroups — #10698540

Order 768.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#73361C211yesmaximal, characteristic
2C3×#1442C211yesmaximal, characteristic
3#71559C212yesmaximal
4C2×#4452C212yesmaximal
5#67613C211yesmaximal, characteristic
6#10407—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#924C2212yes—
9C2×C3×#95—21no—
10C2×C3×#95—21no—
11C2×C3×#95C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5212—21no—
14#5212—21no—
15C2×#830C2212yes—
16C2×#810—22no—
17#4661—22no—
18#4656—22no—
19#4529—22no—
20#4452—24no—
21#2659—31no—
22#1872—32no—
23C2×#328—32no—
24#1442D311yescharacteristic
25#1280—31no—
26C23×C3×C4—21no—
27C23×C3×C4—21no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×Dic6D412yes—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—42no—
34C2×#217—22no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3D412yes—
40#924—22no—
41#924—22no—
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—22no—
45C22×C3⋊C8—22no—
46C3×#95—22no—
47C3×#95—21no—
48C3×#95—42no—
49C3×#95—42no—
50C3×#95D412yes—
51C3⋊#95—22no—
52C3⋊#95—22no—
53#830—22no—
54#830—22no—
55#810—44no—
56C2×C22⋊Q8—32no—
57C22×C4⋊C4C2×D311yescharacteristic
58C2×#103—62no—
59C2×C22⋊C8—32no—
60#398—62no—
61#395—62no—
62#380—61no—
63#380—61no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95C2×D311yescharacteristic
67C2×#95C2×D311yescharacteristic
68#340—62no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4—41no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4SD1612yes—
96C22×C3×C4Q1612yes—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D412yes—
99C22×C3×C4#4111yescharacteristic
100C22×C3×C4#4112yes—
101C22×Dic3—22no—
102C22×Dic3—22no—
103C2×Dic6—22no—
104C2×Dic6—22no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C3×C4⋊C4—24no—
108C3×C4⋊C4—42no—
109#217—42no—
110#217—42no—
111C4⋊Dic3—22no—
112C4⋊Dic3—22no—
113C4⋊Dic3—81no—
114C4⋊Dic3—81no—
115C4⋊Dic3—81no—
116C4⋊Dic3—81no—
117Dic3⋊C4—42no—
118Dic3⋊C4—42no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C22×Q8—32no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—62no—
129C22⋊Q8—62no—
130C22⋊Q8—62no—
131C22⋊Q8—62no—
132C22⋊Q8—62no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C22⋊C4—62no—
142#103—124no—
143C22⋊C8—62no—
144C22⋊C8—62no—
145#95—21no—
146#95—22no—
147#95—42no—
148#95—42no—
149#95—62no—
150#95—62no—
151#95D1212yes—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3#12011yescharacteristic
157C23×C3#12111yescharacteristic
158C23×C3#12411yescharacteristic
159C23×C3C2×SD1611yescharacteristic
160C23×C3C2×Q1611yescharacteristic
161C23×C3#13711yescharacteristic
162C23×C3#13711yescharacteristic
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4#12112yes—
193C2×C3×C4#12312yes—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200Dic6—44no—
201Dic6—82no—
202C3⋊C8—82no—
203C24C2×#6711yescharacteristic
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—122no—
207C2×Q8—122no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4#21412yes—
235C22×C4#21612yes—
236C22×C4C2×D1211yescharacteristic
237C22×C4#23511yescharacteristic
238C22×C4D3×D412yes—
239C22×C4#23912yes—
240C2×C8—122no—
241C2×C8—122no—
242C2×C8—122no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—241no—
250C4⋊C4—241no—
251C4⋊C4—241no—
252C4⋊C4—241no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39411yescharacteristic
268C22×C3#44811yescharacteristic
269C22×C3#45111yescharacteristic
270C22×C3#45411yescharacteristic
271C22×C3#45511yescharacteristic
272C22×C3B(4,𝔽2)11yescharacteristic
273C22×C3#45811yescharacteristic
274C22×C3#46111yescharacteristic
275C22×C3#46211yescharacteristic
276C22×C3#48711yescharacteristic
277C22×C3#49111yescharacteristic
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—42no—
281C3×C4—42no—
282C3×C4—82no—
283C3×C4—82no—
284Dic3—82no—
285Dic3—82no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#93011yescharacteristic
291C23C2×#21411yescharacteristic
292C23#93711yescharacteristic
293C23C2×#21611yescharacteristic
294C23#94711yescharacteristic
295C23#95111yescharacteristic
296C23#95111yescharacteristic
297Q8—124no—
298Q8—242no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4#89412yes—
335C2×C4#89512yes—
336C8—242no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#159911yescharacteristic
342C2×C3#160211yescharacteristic
343C2×C3#160811yescharacteristic
344C2×C3#161711yescharacteristic
345C2×C3#204011yescharacteristic
346C2×C3#204311yescharacteristic
347C2×C3#204411yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22#499111yescharacteristic
361C22#501911yescharacteristic
362C22#503111yescharacteristic
363C22#503211yescharacteristic
364C22#504011yescharacteristic
365C22#505011yescharacteristic
366C22#505311yescharacteristic
367C22#506111yescharacteristic
368C22#506211yescharacteristic
369C22#506911yescharacteristic
370C22#509311yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—82no—
377C4—242no—
378C4—242no—
379C3#1040711yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#6956311yescharacteristic
385C2#6959611yescharacteristic
386C2#6963311yescharacteristic
387C2#6970111yescharacteristic
388C2#6970411yescharacteristic
389C2#6971011yescharacteristic
390C2#6971911yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1442C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#1442D3
hypocenterC3#10407
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1442

Complements

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

NH ≅ G/Ntype
379. C3#10407semidirect

Maximal subgroups

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

subgroupindex
#733612
C3×#14422
#715592
C2×#44522
#676132
#104073