Finite Group Databasebeta

← #10708228

Subgroups — #10708228

Order 768.

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

normal subgroups only

This group has 388 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#76836C211yesmaximal, characteristic
2C3×#1602C211yesmaximal, characteristic
3#71895C211yesmaximal, characteristic
4#71891C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic
6#67451C211yesmaximal, characteristic
7#67447C211yesmaximal, characteristic
8#12254—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C3×#451—21no—
11C3×#393C2211yescharacteristic
12#5230—21no—
13#5044—21no—
14#4949C2211yescharacteristic
15#4946—21no—
16#4946C2211yescharacteristic
17#4944—21no—
18#4942—21no—
19#4658—21no—
20#4652—21no—
21#4647—21no—
22C3×#328—21no—
23C3×#323C2211yescharacteristic
24#4538—21no—
25#4526—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2742—31no—
29#2062—31no—
30#1614—31no—
31#1602D311yescharacteristic
32#1352—31no—
33#1351—31no—
34#1326—31no—
35C22×C3×Q8D411yescharacteristic
36C23×Dic3D411yescharacteristic
37C2×C3×Q16—41no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—41no—
41C3×C22⋊Q8—41no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—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
60#835—41no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—41no—
67C3⋊#95D411yescharacteristic
68#830—21no—
69#807—41no—
70#543C2×D311yescharacteristic
71#484—61no—
72#451—21no—
73#400—61no—
74#399—31no—
75#395—61no—
76#393C2×D311yescharacteristic
77#389—61no—
78#388—61no—
79#387—61no—
80#385—31no—
81#385—61no—
82#380—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—41no—
93C2×C3×Q8—81no—
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—42no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C3×Q16—82no—
108C2×C3×C8—41no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—81no—
117#217—21no—
118#217—22no—
119#217—41no—
120#217—81no—
121#217Q1612yescomplemented
122C4⋊Dic3—41no—
123Dic3⋊C4—42no—
124Dic3⋊C4—42no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127Dic3⋊C4—81no—
128Dic3⋊C4—82no—
129C4×Dic3—81no—
130C2×C3⋊C8—41no—
131C22×Q8#6711yescharacteristic
132C23×C4—31no—
133C2×Q16—41no—
134C22⋊Q8—21no—
135C22⋊Q8—21no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—121no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4C22×D311yescharacteristic
151C2×C42—61no—
152#107—121no—
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—121no—
165#95#6711yescharacteristic
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3#12011yescharacteristic
171C23×C3C2×Q1611yescharacteristic
172C23×C3#13711yescharacteristic
173C3×Q8—42no—
174C3×Q8—81no—
175C3×Q8—81no—
176C3×Q8—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
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—81no—
190C2×C3×C4—81no—
191C2×Dic3—22no—
192C2×Dic3—22no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—42no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C3×C8—81no—
209C3⋊C8—81no—
210C24D3×D411yescharacteristic
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—41no—
214C2×Q8—81no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—122no—
224C22×C4—121no—
225C22×C4—121no—
226C22×C4D3×D411yescharacteristic
227C22×C4C2×#6711yescharacteristic
228Q16—82no—
229C2×C8—41no—
230C2×C8—121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—121no—
236C4⋊C4—122no—
237C4⋊C4—122no—
238C4⋊C4—241no—
239C4⋊C4—241no—
240C4⋊C4—241no—
241C4⋊C4—242no—
242C22⋊C4—32no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—61no—
247C22⋊C4—62no—
248C22⋊C4—81no—
249C22⋊C4—121no—
250C22⋊C4—241no—
251C42—241no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—81no—
261C22×C3#45111yescharacteristic
262C22×C3#45611yescharacteristic
263C22×C3B(4,𝔽2)11yescharacteristic
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4—81no—
270Dic3—42no—
271Dic3—42no—
272Dic3—81no—
273Dic3—82no—
274Dic3—81no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23D3×Q1611yescharacteristic
280C23#91811yescharacteristic
281C23#93711yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—81no—
285Q8—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4—241no—
317C8—81no—
318C8—241no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3#160511yescharacteristic
324C2×C3#219711yescharacteristic
325C2×C3#220011yescharacteristic
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#503211yescharacteristic
336C22#518611yescharacteristic
337C22#520511yescharacteristic
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—122no—
345C4—122no—
346C4—241no—
347C4—242no—
348C4—241no—
349C3#1225411yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2#7098311yescharacteristic
355C2#7099011yescharacteristic
356C2#7099911yescharacteristic
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#12254
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
121. #217Q16semidirect
349. C3#12254semidirect

Maximal subgroups

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

subgroupindex
#768362
C3×#16022
#718952
#718912
#676462
#674512
#674472
#122543