Finite Group Databasebeta

← #10698813

Subgroups — #10698813

Order 768.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×C3×#474C211yesmaximal, characteristic
2#71775C211yesmaximal, characteristic, complemented
3#71774C211yesmaximal, characteristic, complemented
4#71617C211yesmaximal, characteristic
5#71597C211yesmaximal, characteristic
6#70422C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10449—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#474—22no—
12C3×#474—22no—
13C2×C3×#107—21no—
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#102—21no—
16C2×#833—21no—
17#4925C2211yescharacteristic
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4547—22no—
22#4546—22no—
23D4⋊(C3⋊C8)—22no—
24#4530—21no—
25#4479—22no—
26C2×#474D311yescharacteristic
27#1800—31no—
28#1571—31no—
29#1549—31no—
30#1548—31no—
31#1478—31no—
32#1301—31no—
33C22×C3×Q8—21no—
34C22×C3×D4D411yescharacteristic
35C23×C3×C4—21no—
36C22×C3×C8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8D412yes—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52C3×#107—24no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8D412yes—
55C3×#102—24no—
56#833—42no—
57#830—21no—
58#830—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65C4×C3⋊C8—21no—
66C4×C3⋊C8—21no—
67C2×C4⋊Q8C2×D311yescharacteristic
68C2×C4×D4C2×D311yescharacteristic
69#474—22no—
70#474—22no—
71#431—31no—
72#431—31no—
73C2×#107—21no—
74#423—31no—
75#423—31no—
76C2×C4⋊C8C2×D311yescharacteristic
77C2×#102—21no—
78C2×#100—61no—
79#343—61no—
80#335—62no—
81#333—62no—
82#332—62no—
83D4⋊C8—62no—
84C24×C3—21no—
85C2×C3×Q8—22no—
86C2×C3×Q8—41no—
87C2×C3×Q8—82no—
88C2×C3×D4—21no—
89C2×C3×D4—21no—
90C2×C3×D4—21no—
91C2×C3×D4—21no—
92C2×C3×D4—21no—
93C2×C3×D4Q1612yes—
94C22×C3×C4—21no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—82no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#208—41no—
118#208—41no—
119#208—41no—
120#208—41no—
121C2×C3⋊C8—22no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C22×Q8—21no—
126C22×D4#6711yescharacteristic
127C23×C4—21no—
128C2×M16—61no—
129C2×M16—61no—
130C22×C8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8#6712yes—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4#6711yescharacteristic
142C2×C22⋊C4—21no—
143C2×C42C22×D311yescharacteristic
144#107—24no—
145C4⋊C8—21no—
146C4⋊C8—32no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152C4⋊C8#6712yes—
153#102—24no—
154#100—122no—
155C22⋊C8—61no—
156C22⋊C8—61no—
157C4×C8—61no—
158C4×C8—61no—
159C23×C3—22no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—81no—
163C23×C3—81no—
164C23×C3#12311yescharacteristic
165C3×Q8—82no—
166C3×Q8—82no—
167C3×D4—22no—
168C3×D4—22no—
169C3×D4—41no—
170C3×D4—41no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—82no—
176C2×C3×C4—82no—
177C2×C3×C4—82no—
178C2×C3×C4—81no—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12311yescharacteristic
184C2×C3×C4#12411yescharacteristic
185C2×C3×C4C2×Q1611yescharacteristic
186C2×C3×C4#13511yescharacteristic
187C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C3×C8—42no—
190C3⋊C8—42no—
191C3⋊C8—81no—
192C3⋊C8—81no—
193C3⋊C8—81no—
194C24—21no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—82no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4#21612yes—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—82no—
209C22×C4#23711yescharacteristic
210C22×C4#23711yescharacteristic
211C22×C4C2×#6711yescharacteristic
212M16—121no—
213M16—121no—
214M16—121no—
215M16—121no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—62no—
220C2×C8—121no—
221C2×C8—121no—
222C2×C8—121no—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42D3×D411yescharacteristic
232C42#23711yescharacteristic
233C42C2×#6711yescharacteristic
234C42C2×#6711yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—42no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3—81no—
241C22×C3—81no—
242C22×C3#39411yescharacteristic
243C22×C3#48411yescharacteristic
244C22×C3#48511yescharacteristic
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—81no—
249C3×C4—82no—
250C3×C4#44911yescharacteristic
251C3×C4#45111yescharacteristic
252C3×C4#46711yescharacteristic
253C3×C4#46811yescharacteristic
254C3×C4#47212yes—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—81no—
259C23—81no—
260C23#93511yescharacteristic
261Q8—82no—
262Q8—82no—
263D4—22no—
264D4—22no—
265D4—41no—
266D4—41no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4#93511yescharacteristic
276C2×C4#93611yescharacteristic
277C2×C4#93811yescharacteristic
278C2×C4#93911yescharacteristic
279C2×C4#93911yescharacteristic
280C2×C4#94211yescharacteristic
281C2×C4C2×#21611yescharacteristic
282C2×C4#94911yescharacteristic
283C2×C4#95011yescharacteristic
284C2×C4#95311yescharacteristic
285C8—42no—
286C8—122no—
287C8—241no—
288C8—241no—
289C8—241no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—82no—
293C2×C3#203611yescharacteristic
294C2×C3#204511yescharacteristic
295C2×C3#204911yescharacteristic
296C22—21no—
297C22—21no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22#504411yescharacteristic
304C22#504511yescharacteristic
305C22#522211yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—81no—
310C4—82no—
311C4#504911yescharacteristic
312C4#505111yescharacteristic
313C4#516012yes—
314C4#522811yescharacteristic
315C4#523811yescharacteristic
316C3#1044911yescharacteristic, complemented, Sylow
317C2—21no—
318C2—21no—
319C2—82no—
320C2#7169511yescharacteristic
321C2#7204211yescharacteristic
322C2#7210411yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#474C2
socleC22×C3#394
prime radicalC22×C3×D4D4
hypercenterC2×#474D3
hypocenterC3#10449
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. #71775C2semidirect
3. #71774C2semidirect
6. #70422C2semidirect
7. #67707C2semidirect
316. C3#10449semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#717752
#717742
#716172
#715972
#704222
#677072
#104493