Finite Group Databasebeta

← #11780290

Subgroups — #11780290

Order 1152.

528 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 92 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#10593903C211yesmaximal, characteristic
2#10592782C211yesmaximal, characteristic
3C3×#5094C211yesmaximal, characteristic, complemented
4C3×#5080C211yesmaximal, characteristic, complemented
5#10589093C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic
8#71342—31nomaximal
9#71127—31nomaximal
10C2×#3797—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C2×#3755—21no—
13C4×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C3×C4×Dic6C2211yescharacteristic
16#63918—21no—
17#63895—22no—
18#63894—21no—
19#63871—21no—
20#63662—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#810—21no—
23C3×#808—22no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5727—31no—
28Q8×D12—31no—
29C3×#498—31no—
30C3×#475—31no—
31#5094D311yescharacteristic, complemented
32#5083—31no—
33#5083—31no—
34#5083—61no—
35#5080D311yescharacteristic, complemented
36#4547—31no—
37#4535—31no—
38#4483—31no—
39#4482—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×#627—22no—
48C2×#624—22no—
49#3797—42no—
50#3797—42no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C42C2311yescharacteristic
54#3755—42no—
55#3752—21no—
56C4⋊(C3⋊Dic3)D411yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D412yes—
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65Dic3⋊Dic3—22no—
66#3722—22no—
67#1651—91noSylow
68C2×D3×Q8—61no—
69C2×D3×Q8—61no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4×Q8—31no—
75#946—61no—
76C2×#215—61no—
77C2×#215—61no—
78C2×#215—62no—
79#897—61no—
80#897—62no—
81#896—62no—
82#893—61no—
83#888—61no—
84C4×D12—31no—
85C4×D12—31no—
86C4×D12—61no—
87C4×D12—61no—
88#869—61no—
89#869C2×D311yescharacteristic
90C4×Dic6C2×D311yescharacteristic
91C3×#106—61no—
92C3×#106—62no—
93C3×C4⋊C8—31no—
94C3×#103—61no—
95C3×C4×C8—31no—
96#835—61no—
97#835—61no—
98#835—62no—
99#810—21no—
100#808—22no—
101#808—21no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3⋊D3—21no—
108C32×Q8—41no—
109C32×Q8—41no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C3⋊Dic3—21no—
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)SD1612yes—
117C4×C3⋊D3—41no—
118C4×C3⋊D3—41no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—22no—
122C3×Dic6—21no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6SD1612yescomplemented
126#627—44no—
127#624—44no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—41no—
131D4×Q8—91no—
132#498—31no—
133#475—31no—
134#461—91no—
135#332—91no—
136Q8⋊C8—91no—
137D4⋊C8—91no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—31no—
142C2×C3×Q8—61no—
143D3×Q8—122no—
144D3×Q8—122no—
145D3×Q8—122no—
146D3×Q8—122no—
147C2×D12—31no—
148C2×D12—31no—
149C2×D12—61no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×C4×D3—121no—
155C2×Dic6—21no—
156C2×Dic6—62no—
157C2×Dic6—62no—
158C2×Dic6#6711yescharacteristic
159C2×C3×C8—61no—
160C2×C3×C8—61no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C4⋊C4—61no—
170C3×C4⋊C4—61no—
171C3×C42—21no—
172C3×C42—61no—
173C3×C42C22×D311yescharacteristic
174C3×C42C22×D311yescharacteristic
175#215—122no—
176#215—122no—
177#215—124no—
178(C2×D3)⋊C4—62no—
179(C2×D3)⋊C4—62no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—121no—
183C4⋊Dic3—21no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—62no—
188C4⋊Dic3#6711yescharacteristic
189C4⋊Dic3#6712yes—
190Dic3⋊C4—21no—
191Dic3⋊C4—62no—
192C4×Dic3—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—62no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—61no—
198C22×C32#12011yescharacteristic
199C2×C3⋊D3—22no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C32×C4—21no—
203C32×C4—81no—
204C32×C4C2×SD1611yescharacteristic
205C32×C4C2×SD1611yescharacteristic
206C32×C4AGL(1,ℤ/8)11yescharacteristic
207C32×C4#13711yescharacteristic
208C3⋊Dic3—41no—
209C3×Dic3—22no—
210C3×Dic3—42no—
211C3×Dic3—41no—
212C22×Q8—181no—
213C2×SD16—181no—
214C4⋊Q8—31no—
215C4⋊Q8—181no—
216C4⋊Q8D3211yescharacteristic
217C22⋊Q8—182no—
218C22⋊Q8—181no—
219C4×Q8—31no—
220C4×D4—91no—
221C4×D4—181no—
222#106—61no—
223#106—62no—
224C4⋊C8—31no—
225C4⋊C8—91no—
226#103—61no—
227#103—181no—
228C4×C8—31no—
229C22×D3—61no—
230C22×D3—61no—
231C22×D3—121no—
232C3×Q8—32no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—61no—
240C3×Q8—121no—
241C3×Q8—121no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—61no—
249C2×C3×C4—62no—
250C2×C3×C4—61no—
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4D3×D411yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—22no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—121no—
263D12—32no—
264D12—32no—
265D12—62no—
266D12—61no—
267D12—61no—
268D12—121no—
269C4×D3—124no—
270C4×D3—124no—
271C4×D3—121no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—241no—
276C4×D3—241no—
277Dic6—21no—
278Dic6—41no—
279Dic6—41no—
280Dic6—124no—
281Dic6—124no—
282Dic6#21512yescomplemented
283C3×C8—62no—
284C3×C8—121no—
285C3⋊C8—22no—
286C3⋊C8—41no—
287C3⋊C8—122no—
288C3⋊C8—121no—
289C3⋊C8—121no—
290C2×C32#44811yescharacteristic
291C2×C32#45011yescharacteristic
292C2×C32#45611yescharacteristic
293C3⋊D3—42no—
294C2×Q8—21no—
295C2×Q8—31no—
296C2×Q8—61no—
297C2×Q8—182no—
298C2×Q8—362no—
299C2×Q8—362no—
300C2×D4—91no—
301C22×C4—182no—
302C22×C4—181no—
303SD16—362no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—181no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—31no—
310C4⋊C4—32no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—91no—
314C4⋊C4—182no—
315C22⋊C4—182no—
316C22⋊C4—181no—
317C42—61no—
318C42C2×D3211yescharacteristic
319C22×C3—21no—
320C22×C3#93711yescharacteristic
321C22×C3#93711yescharacteristic
322C2×D3—62no—
323C2×D3—62no—
324C2×D3—122no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—241no—
330C2×D3—241no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—41no—
338C3×C4—62no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4—122no—
343C3×C4—121no—
344C3×C4D3×SD1611yescharacteristic
345C3×C4D3×SD1611yescharacteristic
346C3×C4D3×SD1611yescharacteristic
347C3×C4#91411yescharacteristic
348C3×C4#91511yescharacteristic
349C3×C4#91511yescharacteristic
350C3×C4C2×#21511yescharacteristic
351C3×C4#94911yescharacteristic
352Dic3—22no—
353Dic3—42no—
354Dic3—41no—
355Dic3—121no—
356Dic3—121no—
357Dic3—241no—
358C32#165111yescharacteristic, complemented, Sylow
359C23—181no—
360Q8—32no—
361Q8—41no—
362Q8—41no—
363Q8—61no—
364Q8—121no—
365Q8—121no—
366Q8—364no—
367D4—92no—
368D4—181no—
369C2×C4—41no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—62no—
373C2×C4—181no—
374C2×C4—364no—
375C2×C4—361no—
376C2×C4—361no—
377C2×C4D3×#6711yescharacteristic
378C2×C4D3×#6711yescharacteristic
379C2×C4#381111yescharacteristic
380C8—62no—
381C8—121no—
382C8—361no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#504611yescharacteristic
387C2×C3#509511yescharacteristic
388C2×C3#516911yescharacteristic
389C2×C3#517011yescharacteristic
390C2×C3#517011yescharacteristic
391C2×C3#520511yescharacteristic
392D3—122no—
393D3—122no—
394D3—242no—
395C22—182no—
396C22—361no—
397C22—361no—
398C22#6398311yescharacteristic
399C4—21no—
400C4—62no—
401C4—81no—
402C4—122no—
403C4—121no—
404C4—361no—
405C4#6393011yescharacteristic
406C4#6393111yescharacteristic
407C4D3×#21511yescharacteristic
408C4#6393911yescharacteristic
409C3—21no—
410C3#7112711yescharacteristic, complemented
411C3#7134211yescharacteristic, complemented
412C2—362no—
413C2#1059151811yescharacteristic
414C2#1059165111yescharacteristic
415C2#1059166111yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1651
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
3. C3×#5094C2semidirect
4. C3×#5080C2semidirect
5. #10589093C2semidirect
6. #10589084C2semidirect
31. #5094D3semidirect
35. #5080D3semidirect
125. C3×Dic6SD16semidirect
282. Dic6#215semidirect
358. C32#1651semidirect
410. C3#71127semidirect
411. C3#71342semidirect

Maximal subgroups

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

subgroupindex
#105939032
#105927822
C3×#50942
C3×#50802
#105890932
#105890842
#105890822
#713423
#711273