Finite Group Databasebeta

← C4×#64264 #11922876

Subgroups — C4×#64264 #11922876

Order 1152.

704 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#64264C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4#10593628C211yesmaximal, characteristic, complemented
5C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#10593622C211yesmaximal, characteristic, complemented
7C4×C3⋊#226C211yesmaximal, characteristic, complemented
8C4×#983—33nomaximal
9C4×#942—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#64264C414yesdirect product factor
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13C2×C3⋊#226C2211yescharacteristic
14C4×#638—41nomaximal
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3×C4×SL(2,𝔽3)C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3⋊#859C2211yescharacteristic
19C2×#983—33no—
20C2×C3×C4×Q8D311yescharacteristic, complemented
21C2×#942—31no—
22C4×#235—31no—
23C2×C4×SL(2,𝔽3)D313yescomplemented
24#5420—33no—
25#5394—33no—
26C4×GL(2,𝔽3)—33no—
27#5388—33no—
28C4×#226—33no—
29#5225—31no—
30#5030—31no—
31#5030—31no—
32C4×#216—31no—
33C4×#216—31no—
34#5026—31no—
35#5026—31no—
36C4×#215—31no—
37C4×#215—31no—
38#4966—31no—
39#4964—31no—
40C4×#208—31no—
41C22×C32×C4—41no—
42C2×#638—41no—
43C2×C4×C3⋊D3—41no—
44C2×C3×SL(2,𝔽3)C2×C412yescomplemented
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C3⋊GL(2,𝔽3)C2×C412yescomplemented
47C3⋊#226C2×C412yescomplemented
48#3757—41no—
49#3752—41no—
50#3750—41no—
51C4×C3⋊Dic3—41no—
52C4×#137—91noSylow
53C22×C3×Q8C2×D311yescharacteristic
54C2×#235—31no—
55C22×SL(2,𝔽3)C2×D313yes—
56#983—312no—
57C2×GL(2,𝔽3)—33no—
58C2×#226—33no—
59C3×C4×Q8—31no—
60C3×C4×Q8—31no—
61C3×C4×Q8C2×D311yescharacteristic
62C3×C4×Q8C2×D311yescharacteristic
63C2×C3×C4⋊C4—31no—
64C2×C3×C42—31no—
65C2×#216—31no—
66C2×#216—31no—
67#942—34no—
68#942—34no—
69C2×#215—31no—
70C2×#215—31no—
71C4×#67—61no—
72C4×#67—61no—
73C4×#67—123no—
74#926—31no—
75C2×#208—31no—
76C2×#208—31no—
77C4×D12—31no—
78C4×D12—31no—
79#872—61no—
80C42×D3—61no—
81C4×Dic6—31no—
82C4×Dic6—31no—
83C4×SL(2,𝔽3)C2×D313yes—
84C4×SL(2,𝔽3)C2×D313yes—
85#859—33no—
86#859—33no—
87#835—31no—
88#835—31no—
89#835—31no—
90#835—31no—
91#810—31no—
92#810—31no—
93#809—31no—
94#809—31no—
95#808—31no—
96#808—31no—
97#807—31no—
98#807—31no—
99#803—31no—
100#803—31no—
101C4×C3⋊C8—31no—
102C4×C3⋊C8—31no—
103C23×C32—41no—
104C22×C3⋊D3—41no—
105C2×C32×C4—41no—
106C2×C32×C4—41no—
107C2×C32×C4—82no—
108#638—44no—
109C2×C3⋊Dic3—41no—
110C2×C3⋊Dic3—41no—
111C2×C3⋊Dic3—41no—
112C4×C3⋊D3—82no—
113C3×SL(2,𝔽3)C22×C411yescharacteristic
114C2×#137—91no—
115C4×#41—91no—
116C2×C4×Q8C3⋊D311yescharacteristic, complemented
117#441—91no—
118#441—91no—
119#440—91no—
120#440—91no—
121C4×Q16—91no—
122C4×Q16—91no—
123C4×SD16—91no—
124C4×SD16—91no—
125#428—91no—
126#419—91no—
127#417—91no—
128C4×M16—91no—
129C2×C3×Q8—31no—
130C2×C3×Q8—32no—
131C2×C3×Q8C4×D312yescomplemented
132C2×C3×Q8C22×D311yescharacteristic
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—43no—
136C22×C3×C4S411yescharacteristic
137C2×#67—61no—
138C2×#67—123no—
139#235—34no—
140#235—62no—
141C2×D12—31no—
142C2×C4×D3—61no—
143C2×C4×D3—61no—
144C2×C4×D3—61no—
145C2×C4×D3—123no—
146C2×Dic6—31no—
147C2×SL(2,𝔽3)C4×D316yescomplemented
148C2×SL(2,𝔽3)C22×D313yes—
149GL(2,𝔽3)—36no—
150#226—36no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—61no—
154C3×C42—31no—
155C3×C42—31no—
156C3×C42—31no—
157C3×C42—31no—
158#217—61no—
159#217—123no—
160#216—34no—
161#216—32no—
162#216—32no—
163#215—34no—
164#215—32no—
165#215—32no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—123no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171Dic3⋊C4—61no—
172Dic3⋊C4—61no—
173Dic3⋊C4—123no—
174C4×Dic3—61no—
175C4×Dic3—61no—
176C4×Dic3—61no—
177C4×Dic3—123no—
178#208—34no—
179#208—62no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—81no—
187C22×C32—81no—
188C2×C3⋊D3—42no—
189C2×C3⋊D3—81no—
190C2×C3⋊D3—81no—
191C32×C4—42no—
192C32×C4—81no—
193C3⋊Dic3—42no—
194C3⋊Dic3—81no—
195C3⋊Dic3—81no—
196C2×#41—91no—
197C22×Q8C2×C3⋊D311yescharacteristic
198#137—94no—
199#137—94no—
200C2×Q16—91no—
201C2×Q16—91no—
202C2×SD16—91no—
203C2×SD16—91no—
204C2×M16—91no—
205C2×M16—91no—
206C4×Q8—31no—
207C4×Q8—31no—
208C4×Q8—91no—
209C4×Q8—91no—
210C4×Q8C2×C3⋊D311yescharacteristic
211C4×Q8C2×C3⋊D311yescharacteristic
212C4×D4—91no—
213C4×D4—91no—
214C4×D4—181no—
215C4×D4—181no—
216#117—91no—
217#117—181no—
218C2×C4⋊C4—31no—
219C2×C42—31no—
220C2×C42—181no—
221#107—91no—
222#107—91no—
223#106—91no—
224#106—91no—
225#103—91no—
226#103—91no—
227#103—91no—
228#103—91no—
229#103—91no—
230#103—91no—
231#102—91no—
232#102—91no—
233#97—91no—
234#97—91no—
235C4×C8—91no—
236C4×C8—91no—
237C23×C3—43no—
238C23×C3C2×S411yescharacteristic
239C22×D3—61no—
240C22×D3—123no—
241C3×Q8—32no—
242C3×Q8—31no—
243C3×Q8—61no—
244C3×Q8C2×C4×D311yescharacteristic
245C2×C3×C4—22no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—43no—
253C2×C3×C4—43no—
254C2×C3×C4—86no—
255C2×C3×C4—121no—
256C2×C3×C4C2×S411yescharacteristic
257C2×C3×C4C2×S411yescharacteristic
258#67—64no—
259#67—1212no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—123no—
264C2×Dic3—123no—
265C2×Dic3—123no—
266D12—32no—
267D12—61no—
268C4×D3—64no—
269C4×D3—122no—
270C4×D3—122no—
271C4×D3—246no—
272Dic6—32no—
273Dic6—61no—
274SL(2,𝔽3)C2×C4×D313yes—
275C3⋊C8—32no—
276C3⋊C8—32no—
277C3⋊C8—61no—
278C3⋊C8—61no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—82no—
283C3⋊D3—82no—
284#41—94no—
285#41—182no—
286C2×Q8—31no—
287C2×Q8—32no—
288C2×Q8—91no—
289C2×Q8C4×C3⋊D312yescomplemented
290C2×Q8C22×C3⋊D311yescharacteristic
291C2×D4—91no—
292C2×D4—181no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—181no—
296C22×C4—181no—
297C22×C4—181no—
298C22×C4C3⋊S411yescharacteristic
299Q16—94no—
300Q16—92no—
301Q16—92no—
302SD16—94no—
303SD16—92no—
304SD16—92no—
305M16—94no—
306M16—182no—
307C2×C8—91no—
308C2×C8—91no—
309C2×C8—91no—
310C2×C8—91no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—61no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—181no—
317C4⋊C4—181no—
318C22⋊C4—181no—
319C22⋊C4—181no—
320C22⋊C4—181no—
321C42—31no—
322C42—31no—
323C42—31no—
324C42—31no—
325C42—181no—
326C42—181no—
327C42—181no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—46no—
331C22×C3—43no—
332C22×C3—83no—
333C22×C3—83no—
334C22×C3C4×S412yes—
335C22×C3C22×S411yescharacteristic
336C2×D3—62no—
337C2×D3—126no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—243no—
341C2×D3—243no—
342C3×C4—21no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—32no—
346C3×C4—46no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—83no—
350C3×C4#98312yescomplemented
351Dic3—62no—
352Dic3—126no—
353Dic3—121no—
354Dic3—121no—
355Dic3—243no—
356Dic3—243no—
357C32—41noSylow
358C23—181no—
359C23C2×C3⋊S411yescharacteristic
360Q8—32no—
361Q8—31no—
362Q8—61no—
363Q8—92no—
364Q8—181no—
365Q8C2×C4×C3⋊D311yescharacteristic
366D4—92no—
367D4—181no—
368D4—184no—
369C2×C4—22no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—121no—
377C2×C4—184no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—362no—
382C2×C4—362no—
383C2×C4C2×C3⋊S411yescharacteristic
384C2×C4C2×C3⋊S411yescharacteristic
385C8—92no—
386C8—92no—
387C8—181no—
388C8—181no—
389C2×C3—22no—
390C2×C3—43no—
391C2×C3—43no—
392C2×C3—43no—
393C2×C3—86no—
394C2×C3C2×C4×S411yescharacteristic
395C2×C3C2×#98311yescharacteristic
396C2×C3#592311yescharacteristic
397D3—122no—
398D3—246no—
399C22—21no—
400C22—21no—
401C22—182no—
402C22—361no—
403C22—361no—
404C22C4×C3⋊S412yes—
405C22C22×C3⋊S411yescharacteristic
406C4—21no—
407C4—31no—
408C4—31no—
409C4—32no—
410C4—61no—
411C4—61no—
412C4—182no—
413C4—361no—
414C4—361no—
415C4#6426412yesdirect product factor
416C3—43no—
417C3C4×#98311yescharacteristic, complemented
418C2—22no—
419C2—362no—
420C2C2×C4×C3⋊S411yescharacteristic
421C2C2×#6426411yescharacteristic
422C2#1059658611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
prime radicalC2×#64264C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC22×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. #10593654C2semidirect
4. #10593628C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
6. #10593622C2semidirect
7. C4×C3⋊#226C2semidirect
11. #64264C4direct (N × H)
20. C2×C3×C4×Q8D3semidirect
23. C2×C4×SL(2,𝔽3)D3semidirect
44. C2×C3×SL(2,𝔽3)C2×C4semidirect
46. C3⋊GL(2,𝔽3)C2×C4semidirect
47. C3⋊#226C2×C4semidirect
116. C2×C4×Q8C3⋊D3semidirect
131. C2×C3×Q8C4×D3semidirect
147. C2×SL(2,𝔽3)C4×D3semidirect
289. C2×Q8C4×C3⋊D3semidirect
350. C3×C4#983semidirect
415. C4#64264direct (N × H)
417. C3C4×#983semidirect

Maximal subgroups

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

subgroupindex
C2×#642642
C2×C3×C4×SL(2,𝔽3)2
#105936542
#105936282
C4×C3⋊GL(2,𝔽3)2
#105936222
C4×C3⋊#2262
C4×#9833
C4×#9423
C4×#6384