Finite Group Databasebeta

← #11780197

Subgroups — #11780197

Order 1152.

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

normal subgroups only

This group has 522 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#10592786C211yesmaximal, characteristic
3C3×#5090C211yesmaximal, characteristic, complemented
4C3×#5078C211yesmaximal, characteristic, complemented
5#10589093C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic
8#71376—31nomaximal
9#71065—31nomaximal
10C2×#3797—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#216—21no—
14C3×#869C2211yescharacteristic
15C3×C4×Dic6C2211yescharacteristic
16#63918—21no—
17#63895—22no—
18#63894—21no—
19#63871—21no—
20#63636—21no—
21#63634—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×#835—21no—
24C3×#807—22no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5727—31no—
28Q8×D12—31no—
29C3×#500—31no—
30C3×#462—31no—
31#5090D311yescharacteristic, complemented
32#5087—31no—
33#5087—31no—
34#5087—61no—
35#5078D311yescharacteristic, 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#3752—21no—
55C4⋊(C3⋊Dic3)D411yescharacteristic
56C2×C3⋊(C3⋊C8)—21no—
57C3×#216—42no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3D412yes—
60C3×C4⋊Dic3D411yescharacteristic
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#1649—91noSylow
68C2×D3×Q8—61no—
69C2×D3×Q8—61no—
70C2×C3×Q16—61no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—31no—
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4⋊Q8C2×D311yescharacteristic
75C3×C4×Q8—31no—
76#946—61no—
77C2×#216—21no—
78#897—61no—
79#897—62no—
80#896—62no—
81#893—61no—
82#888—61no—
83C4×D12—31no—
84C4×D12—31no—
85C4×D12—61no—
86C4×D12—61no—
87#869—61no—
88#869C2×D311yescharacteristic
89C4×Dic6C2×D311yescharacteristic
90C3×#107—62no—
91C3×C4⋊C8—31no—
92C3×#103—61no—
93C3×C4×C8—31no—
94#835—21no—
95#809—61no—
96#809—61no—
97#809—62no—
98#807—22no—
99#807—61no—
100#807—61no—
101#807—62no—
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)Q1612yescomplemented
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×Dic6D812yes—
126#627—44no—
127#624—44no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—41no—
131D4×Q8—91no—
132#500—31no—
133#474—91no—
134#462—31no—
135#332—91no—
136Q8⋊C8—91no—
137D4⋊C8—91no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—31no—
141C2×C3×Q8—41no—
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
159C3×Q16—122no—
160C2×C3×C8—61no—
161C2×C3×C8—61no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—32no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C42—21no—
173C3×C42—61no—
174C3×C42C22×D311yescharacteristic
175C3×C42C22×D311yescharacteristic
176#216—42no—
177(C2×D3)⋊C4—62no—
178(C2×D3)⋊C4—62no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—121no—
182C4⋊Dic3—21no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—61no—
186C4⋊Dic3—62no—
187C4⋊Dic3#6712yes—
188C4⋊Dic3#6711yescharacteristic
189Dic3⋊C4—21no—
190Dic3⋊C4—62no—
191C4×Dic3—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—62no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—61no—
197C22×C32#12011yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C32×C4—21no—
202C32×C4—81no—
203C32×C4C2×D811yescharacteristic
204C32×C4C2×Q1611yescharacteristic
205C32×C4AGL(1,ℤ/8)11yescharacteristic
206C32×C4#13711yescharacteristic
207C3⋊Dic3—41no—
208C3×Dic3—22no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C22×Q8—181no—
212C2×Q16—61no—
213C4⋊Q8—31no—
214C4⋊Q8—181no—
215C4⋊Q8D3211yescharacteristic
216C22⋊Q8—182no—
217C22⋊Q8—181no—
218C4×Q8—31no—
219C4×D4—91no—
220C4×D4—181no—
221#107—62no—
222#107—181no—
223C4⋊C8—31no—
224C4⋊C8—91no—
225#103—61no—
226#102—181no—
227C4×C8—31no—
228C22×D3—61no—
229C22×D3—61no—
230C22×D3—121no—
231C3×Q8—32no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—61no—
237C3×Q8—81no—
238C3×Q8—81no—
239C3×Q8—121no—
240C3×Q8—121no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—61no—
248C2×C3×C4—62no—
249C2×C3×C4—61no—
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4C2×#6711yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—22no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—121no—
262D12—32no—
263D12—32no—
264D12—62no—
265D12—61no—
266D12—61no—
267D12—121no—
268C4×D3—124no—
269C4×D3—124no—
270C4×D3—121no—
271C4×D3—121no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—241no—
275C4×D3—241no—
276Dic6—21no—
277Dic6—41no—
278Dic6—41no—
279Dic6—124no—
280Dic6—124no—
281Dic6#21312yes—
282C3×C8—62no—
283C3×C8—121no—
284C3⋊C8—22no—
285C3⋊C8—41no—
286C3⋊C8—122no—
287C3⋊C8—121no—
288C3⋊C8—121no—
289C2×C32#44711yescharacteristic
290C2×C32#45111yescharacteristic
291C2×C32#45611yescharacteristic
292C3⋊D3—42no—
293C2×Q8—21no—
294C2×Q8—31no—
295C2×Q8—61no—
296C2×Q8—182no—
297C2×Q8—362no—
298C2×Q8—362no—
299C2×D4—91no—
300C22×C4—182no—
301C22×C4—181no—
302Q16—122no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—181no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—91no—
313C4⋊C4—182no—
314C22⋊C4—182no—
315C22⋊C4—181no—
316C42—61no—
317C42C2×D3211yescharacteristic
318C22×C3—21no—
319C22×C3#93711yescharacteristic
320C22×C3#93711yescharacteristic
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—122no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—241no—
329C2×D3—241no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—41no—
337C3×C4—62no—
338C3×C4—82no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4—122no—
342C3×C4—121no—
343C3×C4D3×D811yescharacteristic
344C3×C4#91111yescharacteristic
345C3×C4D3×Q1611yescharacteristic
346C3×C4D3×Q1611yescharacteristic
347C3×C4#91811yescharacteristic
348C3×C4#91811yescharacteristic
349C3×C4C2×#21311yescharacteristic
350C3×C4#94911yescharacteristic
351Dic3—22no—
352Dic3—42no—
353Dic3—41no—
354Dic3—121no—
355Dic3—121no—
356Dic3—241no—
357C32#164911yescharacteristic, complemented, Sylow
358C23—181no—
359Q8—32no—
360Q8—41no—
361Q8—41no—
362Q8—61no—
363Q8—121no—
364Q8—121no—
365Q8—364no—
366D4—92no—
367D4—181no—
368C2×C4—41no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—62no—
372C2×C4—181no—
373C2×C4—364no—
374C2×C4—361no—
375C2×C4—361no—
376C2×C4D3×#6711yescharacteristic
377C2×C4D3×#6711yescharacteristic
378C2×C4#381111yescharacteristic
379C8—62no—
380C8—121no—
381C8—361no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3#503611yescharacteristic
386C2×C3#509511yescharacteristic
387C2×C3#515611yescharacteristic
388C2×C3#518611yescharacteristic
389C2×C3#518611yescharacteristic
390C2×C3#520511yescharacteristic
391D3—122no—
392D3—122no—
393D3—242no—
394C22—182no—
395C22—361no—
396C22—361no—
397C22#6398311yescharacteristic
398C4—21no—
399C4—62no—
400C4—81no—
401C4—122no—
402C4—121no—
403C4—361no—
404C4D3×#21311yescharacteristic
405C4#6392511yescharacteristic
406C4#6394411yescharacteristic
407C4#6394511yescharacteristic
408C3—21no—
409C3#7106511yescharacteristic, complemented
410C3#7137611yescharacteristic, complemented
411C2—362no—
412C2#1059151511yescharacteristic
413C2#1059165411yescharacteristic
414C2#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#1649
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×#5090C2semidirect
4. C3×#5078C2semidirect
5. #10589093C2semidirect
6. #10589084C2semidirect
31. #5090D3semidirect
35. #5078D3semidirect
116. C4⋊(C3⋊D3)Q16semidirect
357. C32#1649semidirect
409. C3#71065semidirect
410. C3#71376semidirect

Maximal subgroups

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

subgroupindex
#105939032
#105927862
C3×#50902
C3×#50782
#105890932
#105890842
#105890822
#713763
#710653