Finite Group Databasebeta

← #10698532

Subgroups — #10698532

Order 768.

620 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 117 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×#2659C211yesmaximal, characteristic
2#71692C212yesmaximal
3C2×#4538C212yesmaximal
4#69420C211yesmaximal, characteristic
5#67613C211yesmaximal, characteristic
6#10406—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C22×C4⋊Dic3C2211yescharacteristic
9C22×Dic3⋊C4—21no—
10C3×#398—22no—
11C3×#395—22no—
12C3×#380—21no—
13C3×#380—21no—
14C2×C3×#95C2211yescharacteristic
15C2×#835—22no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2212yes—
19#4538—24no—
20#4529—22no—
21#2659D311yescharacteristic
22#1872—32no—
23C2×#328—32no—
24#1444—31no—
25#1280—31no—
26C22×C3×Q8D412yes—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C23×Dic3—21no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×C22⋊Q8—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C22⋊C4—22no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3—42no—
45C2×Dic3⋊C4—42no—
46C2×Dic3⋊C4—42no—
47C2×Dic3⋊C4—42no—
48C22×C3⋊C8—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3×#95—22no—
52#835—44no—
53C3⋊#95—21no—
54C3⋊#95—42no—
55C3⋊#95D412yes—
56#830—22no—
57#830—22no—
58C2×C22⋊Q8C2×D312yes—
59C22×C4⋊C4—31no—
60C22×C4⋊C4—61no—
61C2×#103—62no—
62C2×C22⋊C8—32no—
63#398—22no—
64#395—22no—
65#380—21no—
66#380—21no—
67C2×#95—31no—
68C2×#95—61no—
69C2×#95C2×D311yescharacteristic
70#340—62no—
71#328—64no—
72C24×C3C2×D411yescharacteristic
73C2×C3×Q8—22no—
74C2×C3×Q8—22no—
75C2×C3×Q8—42no—
76C2×C3×Q8—42no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
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×C4SD1612yes—
90C22×C3×C4Q1612yes—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4#4112yes—
94C22×C3×C4#4111yescharacteristic
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—22no—
98C22×Dic3—22no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C22×Dic3—41no—
108C22×Dic3—41no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—81no—
114C3×C4⋊C4—81no—
115C3×C4⋊C4—81no—
116C3×C4⋊C4—81no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119C4⋊Dic3—24no—
120C4⋊Dic3—42no—
121Dic3⋊C4—48no—
122C2×C3⋊C8—42no—
123C2×C3⋊C8—42no—
124C2×C3⋊C8—42no—
125C22×Q8#6712yes—
126C23×C4—61no—
127C23×C4—61no—
128C23×C4#6711yescharacteristic
129C23×C4#6711yescharacteristic
130C23×C4C22×D311yescharacteristic
131C22×C8—62no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—122no—
143C2×C4⋊C4—122no—
144C2×C4⋊C4—122no—
145C2×C4⋊C4—122no—
146C2×C4⋊C4#6712yes—
147C2×C22⋊C4—22no—
148#103—124no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95—32no—
155#95—61no—
156#95—122no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3#12011yescharacteristic
162C23×C3#12111yescharacteristic
163C23×C3#12411yescharacteristic
164C23×C3C2×SD1611yescharacteristic
165C23×C3C2×Q1611yescharacteristic
166C23×C3#13711yescharacteristic
167C23×C3#13711yescharacteristic
168C3×Q8—44no—
169C3×Q8—82no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4#12112yes—
192C2×C3×C4#12312yes—
193C2×Dic3—24no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C3⋊C8—82no—
208C24C2×#6711yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4—121no—
233C22×C4—121no—
234C22×C4—121no—
235C22×C4—121no—
236C22×C4—121no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4#21512yes—
240C22×C4#21612yes—
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4#23712yes—
244C22×C4C2×#6712yes—
245C2×C8—122no—
246C2×C8—122no—
247C2×C8—122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—64no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—122no—
258C4⋊C4—128no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3#39411yescharacteristic
274C22×C3#44811yescharacteristic
275C22×C3#45111yescharacteristic
276C22×C3#45311yescharacteristic
277C22×C3#45611yescharacteristic
278C22×C3B(4,𝔽2)11yescharacteristic
279C22×C3#45811yescharacteristic
280C22×C3#46111yescharacteristic
281C22×C3#46211yescharacteristic
282C22×C3#48711yescharacteristic
283C22×C3#49111yescharacteristic
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—42no—
287C3×C4—42no—
288C3×C4—82no—
289C3×C4—82no—
290Dic3—44no—
291Dic3—82no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#93611yescharacteristic
297C23#93811yescharacteristic
298C23C2×#21511yescharacteristic
299C23#94211yescharacteristic
300C23#94211yescharacteristic
301C23C2×#21611yescharacteristic
302C23#95311yescharacteristic
303Q8—44no—
304Q8—82no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—64no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4#93512yes—
341C2×C4#93912yes—
342C8—242no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#159611yescharacteristic
348C2×C3#160511yescharacteristic
349C2×C3#161111yescharacteristic
350C2×C3#161411yescharacteristic
351C2×C3#203911yescharacteristic
352C2×C3#204311yescharacteristic
353C2×C3#204411yescharacteristic
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22#507911yescharacteristic
367C22#508211yescharacteristic
368C22#508311yescharacteristic
369C22#509011yescharacteristic
370C22#522211yescharacteristic
371C22#522711yescharacteristic
372C22#522811yescharacteristic
373C22#524111yescharacteristic
374C22#524311yescharacteristic
375C22#524411yescharacteristic
376C22#524611yescharacteristic
377C4—22no—
378C4—22no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—82no—
383C4—124no—
384C4—242no—
385C3#1040611yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2#7178111yescharacteristic
391C2#7185011yescharacteristic
392C2#7187611yescharacteristic
393C2#7189111yescharacteristic
394C2#7189211yescharacteristic
395C2#7189811yescharacteristic
396C2#7190411yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2659C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#2659D3
hypocenterC3#10406
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
385. C3#10406semidirect

Maximal subgroups

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

subgroupindex
C3×#26592
#716922
C2×#45382
#694202
#676132
#104063