Finite Group Databasebeta

← #10708219

Subgroups — #10708219

Order 768.

398 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 57 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#76843C211yesmaximal, characteristic
2C3×#1602C211yesmaximal, characteristic
3#71895C211yesmaximal, characteristic
4#71893C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic
6#67449C211yesmaximal, characteristic
7#67447C211yesmaximal, characteristic
8#12253—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C3×#451—21no—
11C3×#393C2211yescharacteristic
12#5229—21no—
13#5050—21no—
14#4950—21no—
15#4947C2211yescharacteristic
16#4946—21no—
17#4946C2211yescharacteristic
18#4944—21no—
19#4659—21no—
20#4656—21no—
21#4650—21no—
22C3×#328—21no—
23C3×#323C2211yescharacteristic
24#4538—21no—
25#4526—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2763—31no—
29#2062—31no—
30#1602—31no—
31#1602D311yescharacteristic
32#1351—31no—
33#1351—31no—
34#1326—31no—
35C22×C3×Q8D411yescharacteristic
36C23×Dic3D411yescharacteristic
37C22×Dic6D411yescharacteristic
38C2×C3×Q16—41no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—41no—
42C3×C22⋊Q8—41no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2311yescharacteristic
46C2×#217—21no—
47C2×#217—21no—
48C2×#217D411yescharacteristic
49C2×#216—41no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—41no—
55C2×C4×Dic3—21no—
56C3×#103—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—41no—
67C3⋊#95D411yescharacteristic
68#830—21no—
69#810—41no—
70#543C2×D311yescharacteristic
71#451—21no—
72#451—61no—
73#400—61no—
74#395—61no—
75#393—31no—
76#393C2×D311yescharacteristic
77#391—61no—
78#388—61no—
79#385—31no—
80#385—61no—
81#385—61no—
82#382—61no—
83#342—31no—
84#342—61no—
85#328—21no—
86#328—61no—
87#323—31no—
88#323C2×D311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×Q8—42no—
93C2×C3×Q8—41no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—81no—
107C2×Dic6—41no—
108C2×Dic6—81no—
109C3×Q16—82no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119#217—21no—
120#217—22no—
121#217—41no—
122#217—81no—
123#217Q1612yescomplemented
124#216—82no—
125C4⋊Dic3—81no—
126C4⋊Dic3—81no—
127Dic3⋊C4—81no—
128Dic3⋊C4—81no—
129Dic3⋊C4—82no—
130C4×Dic3—42no—
131C2×C3⋊C8—41no—
132C22×Q8—31no—
133C22×Q8#6711yescharacteristic
134C23×C4—31no—
135C2×Q16—41no—
136C2×Q16—121no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—121no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4C22×D311yescharacteristic
152C2×C42—61no—
153#103—41no—
154#103—121no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157#95—21no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—121no—
166#95#6711yescharacteristic
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3#12011yescharacteristic
172C23×C3C2×Q1611yescharacteristic
173C23×C3#13711yescharacteristic
174C3×Q8—42no—
175C3×Q8—81no—
176C3×Q8—81no—
177C3×Q8—81no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×Dic3—22no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209Dic6—42no—
210Dic6—81no—
211C3×C8—81no—
212C3⋊C8—81no—
213C24D3×D411yescharacteristic
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—121no—
219C2×Q8—241no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—121no—
229C22×C4—121no—
230C22×C4—241no—
231C22×C4D3×D411yescharacteristic
232C22×C4C2×#6711yescharacteristic
233Q16—82no—
234Q16—242no—
235C2×C8—41no—
236C2×C8—121no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—241no—
242C4⋊C4—241no—
243C4⋊C4—241no—
244C4⋊C4—241no—
245C4⋊C4—242no—
246C22⋊C4—32no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—61no—
252C22⋊C4—62no—
253C22⋊C4—121no—
254C22⋊C4—241no—
255C42—122no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3#45111yescharacteristic
266C22×C3#45611yescharacteristic
267C22×C3B(4,𝔽2)11yescharacteristic
268C3×C4—41no—
269C3×C4—42no—
270C3×C4—41no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—81no—
274Dic3—42no—
275Dic3—81no—
276Dic3—82no—
277Dic3—81no—
278Dic3—81no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23D3×Q1611yescharacteristic
284C23#91811yescharacteristic
285C23#93711yescharacteristic
286Q8—42no—
287Q8—81no—
288Q8—81no—
289Q8—81no—
290Q8—122no—
291Q8—241no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—62no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—241no—
321C2×C4—241no—
322C2×C4—241no—
323C8—81no—
324C8—241no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3#160511yescharacteristic
330C2×C3#219711yescharacteristic
331C2×C3#219911yescharacteristic
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22#503211yescharacteristic
342C22#518611yescharacteristic
343C22#520511yescharacteristic
344C4—41no—
345C4—41no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—122no—
351C4—241no—
352C4—242no—
353C4—241no—
354C4—241no—
355C3#1225311yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2—41no—
359C2—41no—
360C2#7098311yescharacteristic
361C2#7099111yescharacteristic
362C2#7099811yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1602C2
socleC22×C3B(4,𝔽2)
prime radicalC24×C3C2×D4
hypercenter#1602D3
hypocenterC3#12253
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1602

Complements

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

NH ≅ G/Ntype
123. #217Q16semidirect
355. C3#12253semidirect

Maximal subgroups

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

subgroupindex
#768432
C3×#16022
#718952
#718932
#676462
#674492
#674472
#122533