Finite Group Databasebeta

← #11780198

Subgroups — #11780198

Order 1152.

517 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C3×#5727C211yesmaximal, characteristic
2#10592777C211yesmaximal, characteristic, complemented
3#10590385C211yesmaximal, characteristic, complemented
4#10590272C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic
8#71916—31nomaximal
9#68461—31nomaximal
10C2×C3×D3×Q8—21no—
11C32×C4⋊Q8C2211yescharacteristic
12#64078C2211yescharacteristic
13C3×#946—21no—
14C3×#896—22no—
15C3×#893—21no—
16C3×#888—21no—
17C3×C4×D12C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C2×#3719—21no—
20#63634—22no—
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24#63575—21no—
25#63567—21no—
26#63563—21no—
27C3×D4×Q8—31no—
28#5727D311yescharacteristic
29#5078—31no—
30#5078—31no—
31#5078—61no—
32#5031—31no—
33#5007—31no—
34#4849—31no—
35#4846—31no—
36C3×#332—31no—
37#4547D311yescharacteristic, complemented
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4484—31no—
41#4483—31no—
42C2×C32×Q8—21no—
43C2×#634—21no—
44C3×D3×Q8—42no—
45C3×D3×Q8—42no—
46C2×C3×D12D411yescharacteristic
47C2×C3×C4×D3—22no—
48C2×C3×C4×D3—21no—
49C2×C3×Dic6—22no—
50C2×C3×Dic6D411yescharacteristic
51C32×C4⋊C4—21no—
52C32×C4⋊C4D412yes—
53C32×C42C2311yescharacteristic
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C3×(C2×D3)⋊C4—22no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—22no—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#3719—42no—
67#1649—91noSylow
68C22×C3×Q8—61no—
69C2×D3×Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—61no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C22⋊Q8—61no—
75C3×C22⋊Q8—62no—
76C3×C4×Q8—31no—
77C3×C4×D4—31no—
78C3×C4×D4—61no—
79#946—21no—
80C2×#216—61no—
81C2×Dic12—61no—
82#896—22no—
83#893—21no—
84#888—21no—
85C4×D12C2×D311yescharacteristic
86#869—31no—
87#869—31no—
88#869—61no—
89C4×Dic6C2×D311yescharacteristic
90C3×C4⋊C8—31no—
91C3×C4⋊C8—31no—
92#832—61no—
93#821—61no—
94#818—61no—
95#816—61no—
96#810—61no—
97#807—61no—
98#807—64no—
99#807—62no—
100#807—62no—
101C4⋊(C3⋊C8)C2×D311yescharacteristic
102C4⋊(C3⋊C8)C2×D311yescharacteristic
103C4×C3⋊C8—31no—
104C4×C3⋊C8—31no—
105C4×C3⋊C8—61no—
106C22×C3×D3—21no—
107C32×Q8—41no—
108C32×Q8—41no—
109C2×C32×C4—22no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3—41no—
114#634—41no—
115#634—41no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C3×D12—21no—
120C3×D12Q1612yescomplemented
121C3×C4×D3—44no—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×Dic6—21no—
125C3×Dic6—44no—
126C3×Dic6D812yes—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—41no—
129C3×C3⋊C8—41no—
130D4×Q8—31no—
131#500—91no—
132#474—91no—
133#462—91no—
134#332—31no—
135Q8⋊C8—91no—
136D4⋊C8—91no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—31no—
140C2×C3×Q8—41no—
141C2×C3×Q8—62no—
142C2×C3×Q8—122no—
143C2×C3×Q8—122no—
144C2×C3×D4—31no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147D3×Q8—42no—
148D3×Q8—42no—
149C2×D12#6711yescharacteristic
150C2×C4×D3—22no—
151C2×C4×D3—21no—
152C2×Dic6—22no—
153C2×Dic6—61no—
154C2×Dic6—61no—
155C2×Dic6—121no—
156C2×Dic6#6711yescharacteristic
157C2×C3×C8—61no—
158C2×C3×C8—61no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4D1212yes—
168C3×C4⋊C4#6712yes—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—61no—
171C3×C42—21no—
172C3×C42—61no—
173C3×C42C22×D311yescharacteristic
174C3×C42C22×D311yescharacteristic
175#216—122no—
176(C2×D3)⋊C4—22no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3—62no—
179C4⋊Dic3—61no—
180C4⋊Dic3—61no—
181C4⋊Dic3—61no—
182C4⋊Dic3—61no—
183C4⋊Dic3—121no—
184C4⋊Dic3#6711yescharacteristic
185C4⋊Dic3#6711yescharacteristic
186Dic3⋊C4—22no—
187Dic3⋊C4—21no—
188C4×Dic3—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—62no—
192C2×C3⋊C8—61no—
193C2×C3⋊C8—61no—
194Dic12—122no—
195C22×C32#12011yescharacteristic
196C2×C3×D3—22no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C32×C4—21no—
200C32×C4—42no—
201C32×C4C2×D811yescharacteristic
202C32×C4C2×Q1611yescharacteristic
203C32×C4AGL(1,ℤ/8)11yescharacteristic
204C32×C4#13711yescharacteristic
205C3⋊Dic3—81no—
206C3×Dic3—22no—
207C3×Dic3—41no—
208C3×Dic3—41no—
209C22×Q8—61no—
210C2×Q16—181no—
211C4⋊Q8—61no—
212C4⋊Q8—91no—
213C4⋊Q8D3211yescharacteristic
214C22⋊Q8—61no—
215C22⋊Q8—62no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—61no—
219#107—181no—
220#107—182no—
221C4⋊C8—31no—
222C4⋊C8—31no—
223#103—181no—
224#102—181no—
225C4×C8—91no—
226C23×C3—61no—
227C22×D3—21no—
228C3×Q8—32no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—61no—
234C3×Q8—81no—
235C3×Q8—81no—
236C3×Q8—124no—
237C3×D4—32no—
238C3×D4—61no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—42no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—124no—
249C2×C3×C4—121no—
250C2×C3×C4—121no—
251C2×C3×C4C2×D1211yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4D3×D411yescharacteristic
254C2×C3×C4C2×#6711yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—122no—
261C2×Dic3—121no—
262C2×Dic3—121no—
263D12—21no—
264D12#21612yescomplemented
265C4×D3—44no—
266C4×D3—41no—
267C4×D3—41no—
268Dic6—21no—
269Dic6—44no—
270Dic6—121no—
271Dic6—121no—
272Dic6—121no—
273Dic6—121no—
274Dic6—241no—
275Dic6—241no—
276Dic6#21312yes—
277C3×C8—121no—
278C3×C8—121no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—64no—
282C3⋊C8—62no—
283C3⋊C8—62no—
284C2×C32#44711yescharacteristic
285C2×C32#45111yescharacteristic
286C2×C32#45611yescharacteristic
287C3×D3—42no—
288C2×Q8—21no—
289C2×Q8—31no—
290C2×Q8—62no—
291C2×Q8—122no—
292C2×Q8—122no—
293C2×Q8—181no—
294C2×D4—31no—
295C22×C4—62no—
296C22×C4—61no—
297Q16—362no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—181no—
301C4⋊C4—21no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—62no—
305C4⋊C4—61no—
306C4⋊C4—181no—
307C4⋊C4—181no—
308C4⋊C4#62612yes—
309C22⋊C4—62no—
310C22⋊C4—61no—
311C42—61no—
312C42C2×D3211yescharacteristic
313C22×C3—21no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3#88211yescharacteristic
318C22×C3#95311yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C3×C4—22no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—42no—
330C3×C4—42no—
331C3×C4—62no—
332C3×C4—82no—
333C3×C4—121no—
334C3×C4—121no—
335C3×C4D3×D811yescharacteristic
336C3×C4#91411yescharacteristic
337C3×C4#91511yescharacteristic
338C3×C4D3×Q1611yescharacteristic
339C3×C4C2×#21311yescharacteristic
340C3×C4#93211yescharacteristic
341C3×C4#94211yescharacteristic
342C3×C4C2×#21611yescharacteristic
343Dic3—22no—
344Dic3—41no—
345Dic3—41no—
346Dic3—242no—
347Dic3—241no—
348Dic3—241no—
349C32#164911yescharacteristic, complemented, Sylow
350C23—61no—
351Q8—32no—
352Q8—41no—
353Q8—41no—
354Q8—61no—
355Q8—124no—
356Q8—361no—
357Q8—361no—
358D4—32no—
359D4—61no—
360C2×C4—22no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—124no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—361no—
368C2×C4C2×#62611yescharacteristic
369C2×C4D3×#6711yescharacteristic
370C2×C4D3×#6711yescharacteristic
371C8—121no—
372C8—121no—
373C8—182no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—122no—
378C2×C3#475211yescharacteristic
379C2×C3#477311yescharacteristic
380C2×C3#480911yescharacteristic
381C2×C3#521711yescharacteristic
382C2×C3#522811yescharacteristic
383C2×C3#524611yescharacteristic
384D3—42no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22#6397711yescharacteristic
389C4—21no—
390C4—42no—
391C4—62no—
392C4—121no—
393C4—121no—
394C4—721no—
395C4D3×#21311yescharacteristic
396C4#6393411yescharacteristic
397C4D3×#21611yescharacteristic
398C4#6394611yescharacteristic
399C3—21no—
400C3#6846111yescharacteristic, complemented
401C3#7191611yescharacteristic, complemented
402C2—122no—
403C2#1059144811yescharacteristic
404C2#1059159511yescharacteristic
405C2#1059164911yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
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
2. #10592777C2semidirect
3. #10590385C2semidirect
5. C3×#4547C2semidirect
6. #10589083C2semidirect
37. #4547D3semidirect
120. C3×D12Q16semidirect
264. D12#216semidirect
349. C32#1649semidirect
400. C3#68461semidirect
401. C3#71916semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
#105927772
#105903852
#105902722
C3×#45472
#105890832
#105890802
#719163
#684613