Finite Group Databasebeta

← #10698537

Subgroups — #10698537

Order 768.

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

normal subgroups only

This group has 576 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#69415C211yesmaximal, characteristic
5#67613C211yesmaximal, characteristic
6#10407—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C22×C4⋊Dic3C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×#380—21no—
12C3×#380—21no—
13C2×C3×#95C2211yescharacteristic
14C2×#835—22no—
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2212yes—
19#4538—24no—
20#4529—22no—
21#2659D311yescharacteristic
22#1872—32no—
23C2×#328—32no—
24#1442—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—
45C22×C3⋊C8—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—22no—
49#835—44no—
50C3⋊#95—21no—
51C3⋊#95—42no—
52C3⋊#95—42no—
53C3⋊#95D412yes—
54#830—22no—
55#830—22no—
56C2×C22⋊Q8C2×D312yes—
57C22×C4⋊C4—31no—
58C2×#103—62no—
59C2×C22⋊C8—32no—
60#398—22no—
61#395—22no—
62#380—21no—
63#380—21no—
64C2×#95—31no—
65C2×#95—61no—
66C2×#95—61no—
67C2×#95C2×D311yescharacteristic
68#340—62no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—42no—
74C2×C3×Q8—42no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
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×C4SD1612yes—
88C22×C3×C4Q1612yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4112yes—
92C22×C3×C4#4111yescharacteristic
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—81no—
112C3×C4⋊C4—81no—
113C3×C4⋊C4—81no—
114C3×C4⋊C4—81no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117C4⋊Dic3—24no—
118C4⋊Dic3—42no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C22×Q8#6712yes—
123C23×C4—61no—
124C23×C4—61no—
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—62no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—122no—
140C2×C4⋊C4#6712yes—
141C2×C22⋊C4—22no—
142#103—124no—
143C22⋊C8—62no—
144C22⋊C8—62no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—32no—
149#95—61no—
150#95—122no—
151#95—122no—
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
163C3×Q8—44no—
164C3×Q8—82no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
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—42no—
183C2×C3×C4—42no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4#12112yes—
187C2×C3×C4#12312yes—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C3⋊C8—82no—
203C24C2×#6711yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—42no—
207C2×Q8—42no—
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—62no—
221C22×C4—62no—
222C22×C4—121no—
223C22×C4—121no—
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—121no—
233C22×C4—121no—
234C22×C4#21512yes—
235C22×C4#21612yes—
236C22×C4D3×D411yescharacteristic
237C22×C4#23711yescharacteristic
238C22×C4#23712yes—
239C22×C4C2×#6712yes—
240C2×C8—122no—
241C2×C8—122no—
242C2×C8—122no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—64no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—122no—
253C22⋊C4—42no—
254C22⋊C4—42no—
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#93611yescharacteristic
291C23#93811yescharacteristic
292C23C2×#21511yescharacteristic
293C23#94211yescharacteristic
294C23#94211yescharacteristic
295C23C2×#21611yescharacteristic
296C23#95311yescharacteristic
297Q8—44no—
298Q8—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
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—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#93512yes—
335C2×C4#93912yes—
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#507911yescharacteristic
361C22#508211yescharacteristic
362C22#508311yescharacteristic
363C22#509011yescharacteristic
364C22#522211yescharacteristic
365C22#522711yescharacteristic
366C22#522811yescharacteristic
367C22#524211yescharacteristic
368C22#524311yescharacteristic
369C22#524411yescharacteristic
370C22#524511yescharacteristic
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#7178111yescharacteristic
385C2#7185011yescharacteristic
386C2#7187711yescharacteristic
387C2#7189011yescharacteristic
388C2#7189311yescharacteristic
389C2#7189911yescharacteristic
390C2#7190211yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2659C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#2659D3
hypocenterC3#10407
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
379. C3#10407semidirect

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
#694152
#676132
#104073