Finite Group Databasebeta

← #11780246

Subgroups — #11780246

Order 1152.

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

normal subgroups only

This group has 506 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#10596203C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590379C211yesmaximal, characteristic
4#10590307C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589085C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic
8#71370—31nomaximal
9#68437—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64156—21no—
12Q8×C3⋊Dic3—21no—
13#64107—22no—
14#64106—21no—
15#64101—21no—
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C4×#634C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C2×#3717—21no—
20C3×#807—22no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63573—21no—
25#63567—21no—
26#63566—21no—
27#5732—31no—
28#5732—31no—
29#5732—61no—
30C3×#500—31no—
31#5094—31no—
32#5083—31no—
33#5078D311yescharacteristic, complemented
34#4851—31no—
35#4841—31no—
36C3×#332—31no—
37#4547D311yescharacteristic, complemented
38#4484—31no—
39#4483—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×#634D411yescharacteristic
46C2×C3×Dic6—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4D412yes—
49C32×C42C2311yescharacteristic
50#3752—22no—
51#3752—22no—
52#3752—21no—
53C4⋊(C3⋊Dic3)—22no—
54C4⋊(C3⋊Dic3)D411yescharacteristic
55C4⋊(C3⋊Dic3)D411yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—22no—
58C4×C3⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—21no—
64#3717—42no—
65#1650—91noSylow
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—31no—
68C3×C4⋊Q8C2×D311yescharacteristic
69C3×C4⋊Q8C2×D311yescharacteristic
70#947—61no—
71#947—61no—
72#947—62no—
73Q8×Dic3—61no—
74Q8×Dic3—61no—
75Q8×Dic3—62no—
76C2×#215—61no—
77C2×#204—61no—
78#898—64no—
79#898—62no—
80#898—62no—
81#897—61no—
82#897—61no—
83#897—62no—
84#892—61no—
85#892—61no—
86#892—62no—
87C4×D12—31no—
88C4×D12—31no—
89C4×D12—61no—
90#869C2×D311yescharacteristic
91C4×Dic6—31no—
92C4×Dic6—31no—
93C4×Dic6—61no—
94C3×#107—62no—
95C3×C4⋊C8—31no—
96C3×C4⋊C8—31no—
97C3×C4×C8—31no—
98#835—61no—
99#817—61no—
100#816—61no—
101#816—61no—
102#810—61no—
103#808—61no—
104#807—22no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4⋊(C3⋊C8)C2×D311yescharacteristic
107C4×C3⋊C8C2×D311yescharacteristic
108C22×C3⋊D3—21no—
109C32×Q8—41no—
110C32×Q8—41no—
111C2×C32×C4—22no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)—21no—
120C4×C3⋊D3—41no—
121C4×C3⋊D3—41no—
122#634—21no—
123#634—21no—
124C2×C3×Dic3—41no—
125C3×Dic6—41no—
126C3×Dic6—41no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—41no—
129C3×C3⋊C8—41no—
130#556—91no—
131#500—31no—
132#475—91no—
133#461—91no—
134#332—31no—
135Q8⋊C8—91no—
136D4⋊C8—91no—
137C2×C3×Q8—22no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—61no—
141C2×D12—31no—
142C2×D12—31no—
143C2×D12—61no—
144C2×C4×D3—62no—
145C2×C4×D3—61no—
146C2×C4×D3—61no—
147C2×Dic6—21no—
148C2×Dic6—31no—
149C2×Dic6—31no—
150C2×Dic6—61no—
151C2×C3×C8—61no—
152C2×C3×C8—61no—
153C2×C3×C8—61no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—61no—
160C3×C4⋊C4D1212yes—
161C3×C4⋊C4#6712yes—
162C3×C42—21no—
163C3×C42C22×D311yescharacteristic
164C3×C42C22×D311yescharacteristic
165#215—122no—
166(C2×D3)⋊C4—62no—
167(C2×D3)⋊C4—64no—
168(C2×D3)⋊C4—62no—
169(C2×D3)⋊C4—62no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—62no—
172(C2×D3)⋊C4—62no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—61no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—61no—
182C4⋊Dic3—61no—
183C4⋊Dic3—64no—
184C4⋊Dic3—62no—
185C4⋊Dic3—62no—
186Dic3⋊C4—62no—
187Dic3⋊C4—61no—
188Dic3⋊C4—61no—
189C4×Dic3—62no—
190C4×Dic3—64no—
191C4×Dic3—62no—
192C4×Dic3—62no—
193C4×Dic3—61no—
194C4×Dic3—61no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198#204—122no—
199C22×C32#12011yescharacteristic
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C32×C4—21no—
204C32×C4—42no—
205C32×C4#13511yescharacteristic
206C32×C4#13511yescharacteristic
207C32×C4AGL(1,ℤ/8)11yescharacteristic
208C32×C4#13711yescharacteristic
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3⋊Dic3—41no—
212C3×Dic3—81no—
213C2×SD16—181no—
214C4⋊Q8—31no—
215C4⋊Q8D3211yescharacteristic
216#126—182no—
217#124—181no—
218C22⋊Q8—181no—
219C4×Q8—91no—
220C4×Q8—181no—
221C4×D4—91no—
222#117—181no—
223#107—62no—
224#106—181no—
225C4⋊C8—31no—
226C4⋊C8—31no—
227#103—181no—
228#103—181no—
229C4×C8—31no—
230C22×D3—62no—
231C22×D3—61no—
232C22×D3—61no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—121no—
240C3×Q8—121no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—121no—
248C2×C3×C4C2×D1211yescharacteristic
249C2×C3×C4D3×D411yescharacteristic
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4C2×#6711yescharacteristic
254C2×Dic3—41no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264D12—61no—
265D12—61no—
266D12—61no—
267D12—61no—
268D12—121no—
269D12—121no—
270C4×D3—122no—
271C4×D3—122no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—121no—
276Dic6—41no—
277Dic6—41no—
278Dic6—61no—
279Dic6—61no—
280Dic6—61no—
281Dic6—61no—
282Dic6—121no—
283Dic6—121no—
284C3×C8—62no—
285C3×C8—121no—
286C3×C8—121no—
287C3⋊C8—22no—
288C3⋊C8—41no—
289C3⋊C8—41no—
290C2×C32#44911yescharacteristic
291C2×C32#45211yescharacteristic
292C2×C32#45611yescharacteristic
293C3⋊D3—81no—
294C2×Q8—21no—
295C2×Q8—61no—
296C2×Q8—91no—
297C2×D4—91no—
298C22×C4—181no—
299SD16—362no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—61no—
303C4⋊C4—21no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—182no—
309C4⋊C4—181no—
310C4⋊C4#62612yes—
311C22⋊C4—182no—
312C22⋊C4—182no—
313C22⋊C4—181no—
314C42—182no—
315C42—181no—
316C42C2×D3211yescharacteristic
317C22×C3—21no—
318C22×C3#88211yescharacteristic
319C22×C3#93711yescharacteristic
320C2×D3—122no—
321C2×D3—122no—
322C2×D3—122no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—41no—
336C3×C4—44no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—241no—
340C3×C4#91111yescharacteristic
341C3×C4#91111yescharacteristic
342C3×C4#91211yescharacteristic
343C3×C4#91611yescharacteristic
344C3×C4#91611yescharacteristic
345C3×C4#91811yescharacteristic
346C3×C4#91811yescharacteristic
347C3×C4#91911yescharacteristic
348Dic3—81no—
349Dic3—122no—
350Dic3—122no—
351Dic3—122no—
352Dic3—121no—
353Dic3—121no—
354Dic3—121no—
355Dic3—121no—
356Dic3—121no—
357Dic3—121no—
358C32#165011yescharacteristic, complemented, Sylow
359C23—181no—
360Q8—41no—
361Q8—41no—
362Q8—121no—
363Q8—121no—
364Q8—181no—
365Q8—181no—
366D4—181no—
367D4—181no—
368C2×C4—22no—
369C2×C4—121no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—361no—
374C2×C4—361no—
375C2×C4C2×#62611yescharacteristic
376C2×C4#381111yescharacteristic
377C2×C4#381111yescharacteristic
378C8—62no—
379C8—121no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3#475211yescharacteristic
385C2×C3#478111yescharacteristic
386C2×C3#480811yescharacteristic
387C2×C3#509511yescharacteristic
388C2×C3#515811yescharacteristic
389C2×C3#518711yescharacteristic
390D3—242no—
391D3—241no—
392D3—241no—
393C22—361no—
394C22—361no—
395C22—361no—
396C22#6398111yescharacteristic
397C4—21no—
398C4—42no—
399C4—241no—
400C4—361no—
401C4—361no—
402C4—361no—
403C4#6392711yescharacteristic
404C4#6393611yescharacteristic
405C4#6394511yescharacteristic
406C4#6395011yescharacteristic
407C3—21no—
408C3#6843711yescharacteristic, complemented
409C3#7137011yescharacteristic, complemented
410C2—721no—
411C2#1059145711yescharacteristic
412C2#1059160211yescharacteristic
413C2#1059165011yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1650
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×#5078C2semidirect
4. #10590307C2semidirect
5. C3×#4547C2semidirect
6. #10589085C2semidirect
33. #5078D3semidirect
37. #4547D3semidirect
358. C32#1650semidirect
408. C3#68437semidirect
409. C3#71370semidirect

Maximal subgroups

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

subgroupindex
#105962032
C3×#50782
#105903792
#105903072
C3×#45472
#105890852
#105890822
#713703
#684373