Finite Group Databasebeta

← #11780954

Subgroups — #11780954

Order 1152.

398 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 88 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#10592786C211yesmaximal, characteristic
2C3×#5092C211yesmaximal, characteristic, complemented
3#10590327C211yesmaximal, characteristic
4#10590313C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71305—31nomaximal
9#68441—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#216—21no—
13C2×C3×#216—21no—
14C3×#869C2211yescharacteristic
15C2×#3710—21no—
16#63636—21no—
17#63634—21no—
18C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63575—21no—
23#63564—21no—
24#63557—21no—
25C3×#494—31no—
26#5092D311yescharacteristic, complemented
27#5087—31no—
28#5087—31no—
29#5087—31no—
30#5087—61no—
31#4846—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#332—31no—
35#4547—31no—
36#4547D311yescharacteristic, complemented
37#4482—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)D411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3752—21no—
48C4⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
50C3×#216—42no—
51C3×#216—42no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#3710—42no—
59#1669—91noSylow
60C2×C3×Q16—61no—
61C2×C3×Q16—61no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—31no—
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×C4⋊Q8C2×D311yescharacteristic
66C2×#216—21no—
67C2×#216—21no—
68C2×D3×C8—61no—
69C2×D3×C8—61no—
70C4×D12—31no—
71C4×D12—31no—
72C4×D12—61no—
73#869C2×D311yescharacteristic
74C3×C4⋊C8—31no—
75C3×C4⋊C8—31no—
76C3×C4×C8—31no—
77#821—61no—
78(C2×D3)⋊C8—61no—
79(C2×D3)⋊C8—61no—
80#818—61no—
81#809—61no—
82#809—61no—
83#809—61no—
84#809—62no—
85#807—61no—
86#807—61no—
87#807—61no—
88#807—62no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—61no—
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4⋊(C3⋊C8)C2×D311yescharacteristic
94C4×C3⋊C8C2×D311yescharacteristic
95C22×C3⋊D3—21no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C3⋊Dic3—21no—
103C4⋊(C3⋊D3)—21no—
104C4⋊(C3⋊D3)Q1612yescomplemented
105C4×C3⋊D3—41no—
106C4×C3⋊D3—41no—
107C2×C3×Dic3—41no—
108C3×Dic6—41no—
109C3×Dic6—41no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8D812yes—
115#494—31no—
116#474—91no—
117#474—91no—
118C8×D4—91no—
119#335—91no—
120#332—31no—
121#332—91no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—41no—
125C2×C3×Q8—61no—
126C2×D12—31no—
127C2×D12—31no—
128C2×D12—61no—
129C2×C4×D3—61no—
130C2×C4×D3—61no—
131C2×C4×D3—121no—
132C2×Dic6—21no—
133C3×Q16—122no—
134C3×Q16—122no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—61no—
138C2×C3×C8—61no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—61no—
147C3×C42—21no—
148C3×C42C22×D311yescharacteristic
149C3×C42C22×D311yescharacteristic
150#216—42no—
151#216—42no—
152(C2×D3)⋊C4—61no—
153(C2×D3)⋊C4—61no—
154(C2×D3)⋊C4—121no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—62no—
163C2×C3⋊C8—61no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8#6711yescharacteristic
166C2×C3⋊C8#6711yescharacteristic
167D3×C8—122no—
168D3×C8—122no—
169C22×C32#12111yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—41no—
173C32×C4—21no—
174C32×C4—81no—
175C32×C4C2×D811yescharacteristic
176C32×C4C2×Q1611yescharacteristic
177C32×C4#13511yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C3⋊Dic3—41no—
180C3×Dic3—81no—
181C2×Q16—61no—
182C2×Q16—61no—
183C22×C8—181no—
184C4⋊Q8—31no—
185C4⋊Q8D3211yescharacteristic
186C4×D4—91no—
187#107—181no—
188#107—181no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—91no—
192#102—181no—
193#102—181no—
194C22⋊C8—181no—
195C4×C8—31no—
196C22×D3—61no—
197C22×D3—61no—
198C22×D3—121no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—81no—
204C3×Q8—81no—
205C3×Q8—121no—
206C3×Q8—121no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—121no—
214C2×C3×C4#23511yescharacteristic
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4D3×D411yescharacteristic
218C2×C3×C4#23711yescharacteristic
219C2×C3×C4C2×#6711yescharacteristic
220C2×Dic3—41no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—121no—
224D12—32no—
225D12—32no—
226D12—62no—
227D12—61no—
228D12—61no—
229D12—121no—
230C4×D3—121no—
231C4×D3—121no—
232C4×D3—121no—
233C4×D3—121no—
234C4×D3—241no—
235C4×D3—241no—
236Dic6—41no—
237Dic6—41no—
238C3×C8—32no—
239C3×C8—61no—
240C3×C8—121no—
241C3×C8—121no—
242C3⋊C8—21no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C3⋊C8—122no—
246C3⋊C8—121no—
247C3⋊C8—121no—
248C3⋊C8#21312yes—
249C2×C32#45911yescharacteristic
250C2×C32#46711yescharacteristic
251C2×C32#47311yescharacteristic
252C3⋊D3—42no—
253C2×Q8—21no—
254C2×Q8—61no—
255C2×D4—91no—
256C22×C4—181no—
257Q16—122no—
258Q16—122no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—181no—
264C2×C8—362no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—91no—
270C22⋊C4—181no—
271C42C2×D3211yescharacteristic
272C22×C3—21no—
273C22×C3#88411yescharacteristic
274C22×C3#93811yescharacteristic
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—122no—
278C2×D3—121no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—241no—
283C2×D3—241no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—82no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4—241no—
295C3×C4D3×D811yescharacteristic
296C3×C4#91411yescharacteristic
297C3×C4#91611yescharacteristic
298C3×C4D3×Q1611yescharacteristic
299C3×C4D3×Q1611yescharacteristic
300C3×C4#91911yescharacteristic
301C3×C4C2×#21311yescharacteristic
302C3×C4#93211yescharacteristic
303Dic3—81no—
304Dic3—121no—
305Dic3—121no—
306Dic3—241no—
307C32#166911yescharacteristic, complemented, Sylow
308C23—181no—
309Q8—41no—
310Q8—41no—
311Q8—121no—
312Q8—121no—
313D4—92no—
314D4—181no—
315C2×C4—41no—
316C2×C4—121no—
317C2×C4—181no—
318C2×C4—361no—
319C2×C4—361no—
320C2×C4#380411yescharacteristic
321C2×C4D3×#6711yescharacteristic
322C2×C4#381111yescharacteristic
323C8—32no—
324C8—61no—
325C8—121no—
326C8—121no—
327C8—361no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#475511yescharacteristic
332C2×C3#478611yescharacteristic
333C2×C3#481311yescharacteristic
334C2×C3#507211yescharacteristic
335C2×C3#518811yescharacteristic
336C2×C3#519211yescharacteristic
337D3—122no—
338D3—122no—
339D3—242no—
340C22—182no—
341C22—361no—
342C22—361no—
343C22#6395811yescharacteristic
344C4—21no—
345C4—81no—
346C4—241no—
347C4—361no—
348C4D3×#21311yescharacteristic
349C4#6393411yescharacteristic
350C4#6394411yescharacteristic
351C4#6395011yescharacteristic
352C3—21no—
353C3#6844111yescharacteristic, complemented
354C3#7130511yescharacteristic, complemented
355C2—362no—
356C2#1059142911yescharacteristic
357C2#1059154311yescharacteristic
358C2#1059160111yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1669
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5092C2semidirect
5. C3×#4547C2semidirect
6. #10589099C2semidirect
7. #10589093C2semidirect
26. #5092D3semidirect
36. #4547D3semidirect
104. C4⋊(C3⋊D3)Q16semidirect
307. C32#1669semidirect
353. C3#68441semidirect
354. C3#71305semidirect

Maximal subgroups

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

subgroupindex
#105927862
C3×#50922
#105903272
#105903132
C3×#45472
#105890992
#105890932
#713053
#684413