Finite Group Databasebeta

← C22×#63804 #11918476

Subgroups — C22×#63804 #11918476

Order 1152.

3157 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3798C211yesmaximal, characteristic
2C22×#3743C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C22×C3×D3×C8C211yesmaximal, characteristic, complemented
5C22×#3711C211yesmaximal, characteristic, complemented
6C22×#3710C211yesmaximal, characteristic, complemented
7C22×D3×C3⋊C8C211yesmaximal, characteristic, complemented
8C2×#63804C2124yesmaximal, direct product factor
9C22×#907—31nomaximal
10C22×#901—31nomaximal
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×C3×C4×D3C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×#627C411yescharacteristic
15C22×#626C411yescharacteristic
16C22×#626C411yescharacteristic
17C22×#625C411yescharacteristic
18C2×#3798C4112yes—
19C2×#3798C22112yes—
20C22×C32×C8C2211yescharacteristic
21C22×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×#3743C22112yescomplemented
23C22×C3×C3⋊C8C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C2×C3×ΓL(1,𝔽25)C22112yescomplemented
26C2×C3×D3×C8C22112yescomplemented
27C2×#3711C22112yescomplemented
28C2×#3710C22112yescomplemented
29C2×D3×C3⋊C8C22112yescomplemented
30#63804C22164yesdirect product factor
31C22×#235—31no—
32C22×#235—31no—
33C22×C3×M16—31no—
34C23×C3×C8—31no—
35C22×#208—31no—
36C23×C3⋊C8—31no—
37C2×#907—324no—
38C2×#901—324no—
39C22×ΓL(1,𝔽25)—31no—
40C22×ΓL(1,𝔽25)—31no—
41C22×ΓL(1,𝔽25)—31no—
42C22×ΓL(1,𝔽25)—61no—
43C22×ΓL(1,𝔽25)D311yescharacteristic, complemented
44C22×D3×C8—31no—
45C22×D3×C8—31no—
46C22×D3×C8—31no—
47C22×D3×C8D311yescharacteristic, complemented
48C23×C3⋊D3C2×C411yescharacteristic
49C23×C3×D3C2×C411yescharacteristic
50C23×C3×D3C2×C411yescharacteristic
51C22×C32×C4C2311yescharacteristic
52C22×C3⋊Dic3C2×C411yescharacteristic
53C2×C4×C3⋊D3C2×C416yes—
54C2×C4×C3⋊D3C2316yes—
55C22×C3×Dic3C2×C411yescharacteristic
56C22×C3×Dic3C2×C411yescharacteristic
57C2×C3×C4×D3C2×C416yes—
58C2×C3×C4×D3C2×C416yes—
59C2×C3×C4×D3C2316yes—
60C2×C3×C4×D3C2316yes—
61C2×#627C2×C4112yes—
62C2×#626C2×C4112yes—
63C2×#626C2×C4112yes—
64C2×#625C2×C4112yes—
65#3798C2×C4148yes—
66#3798C23116yes—
67C2×C32×C8C2316yes—
68C2×C3⋊(C3⋊C8)C2316yes—
69#3743C23116yescomplemented
70C2×C3×C3⋊C8C2316yes—
71C2×C3×C3⋊C8C2316yes—
72C3×ΓL(1,𝔽25)C23116yescomplemented
73C3×D3×C8C23116yescomplemented
74#3711C23116yescomplemented
75#3710C23116yescomplemented
76D3×C3⋊C8C23116yescomplemented
77C22×#131—91noSylow
78C23×C3×C4—31no—
79C23×C3×C4—31no—
80C22×#67—31no—
81C22×#67—31no—
82C22×#67—31no—
83C22×#67—31no—
84C2×#235—312no—
85C2×#235—312no—
86C2×#235—312no—
87C2×#235—312no—
88C22×D12—31no—
89C22×D12—31no—
90C22×C4×D3—31no—
91C22×C4×D3—31no—
92C22×C4×D3—61no—
93C22×C4×D3C2×D311yescharacteristic
94C22×C4×D3C2×D311yescharacteristic
95C22×Dic6—31no—
96C22×Dic6—31no—
97C2×C3×M16—312no—
98C22×C3×C8—21no—
99C22×C3×C8—31no—
100C22×C3×C8—31no—
101C22×C3×C8—312no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C2×D311yescharacteristic
104C2×#208—312no—
105C22×C3⋊C8—31no—
106C22×C3⋊C8—31no—
107C22×C3⋊C8—312no—
108C22×C3⋊C8—61no—
109C22×C3⋊C8C2×D311yescharacteristic
110C22×C3⋊C8C2×D311yescharacteristic
111#907—364no—
112#901—364no—
113C2×ΓL(1,𝔽25)—312no—
114C2×ΓL(1,𝔽25)—312no—
115C2×ΓL(1,𝔽25)—312no—
116C2×ΓL(1,𝔽25)—612no—
117C2×ΓL(1,𝔽25)C2×D3112yescomplemented
118C2×D3×C8—312no—
119C2×D3×C8—312no—
120C2×D3×C8—312no—
121C2×D3×C8C2×D3112yescomplemented
122C23×C32C22×C411yescharacteristic
123C22×C3⋊D3—24no—
124C22×C3⋊D3C22×C416yes—
125C22×C3×D3—24no—
126C22×C3×D3—24no—
127C22×C3×D3C22×C416yes—
128C22×C3×D3C22×C416yes—
129C2×C32×C4C22×C413yes—
130C2×C32×C4C2413yes—
131C2×C3⋊Dic3C22×C416yes—
132C4×C3⋊D3C22×C4112yes—
133C4×C3⋊D3C2414yes—
134C2×C3×Dic3C22×C416yes—
135C2×C3×Dic3C22×C416yes—
136C3×C4×D3C22×C4112yes—
137C3×C4×D3C22×C4112yes—
138C3×C4×D3C2414yes—
139C3×C4×D3C2414yes—
140#627C22×C4116yes—
141#626C22×C4116yes—
142#626C22×C4116yes—
143#625C22×C4116yes—
144C32×C8C2414yes—
145C3⋊(C3⋊C8)C2414yes—
146C3×C3⋊C8C2414yes—
147C3×C3⋊C8C2414yes—
148C22×#41—91no—
149C2×#131—924no—
150C22×M16—31no—
151C22×M16—91no—
152C22×M16—91no—
153C23×C8—31no—
154C23×C8—91no—
155C23×C8—91no—
156C24×C3—31no—
157C24×C3—31no—
158C23×D3—31no—
159C23×D3—31no—
160C23×D3—61no—
161C23×D3C4×D311yescharacteristic
162C23×D3C4×D311yescharacteristic
163C22×C3×C4—21no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—36no—
167C22×C3×C4—36no—
168C22×C3×C4—36no—
169C22×C3×C4—36no—
170C22×C3×C4C22×D311yescharacteristic
171C22×C3×C4C22×D311yescharacteristic
172C2×#67—312no—
173C2×#67—312no—
174C2×#67—312no—
175C2×#67—312no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—61no—
179C22×Dic3C4×D311yescharacteristic
180C22×Dic3C4×D311yescharacteristic
181#235—348no—
182#235—348no—
183#235—316no—
184#235—316no—
185C2×D12—312no—
186C2×D12—312no—
187C2×C4×D3—36no—
188C2×C4×D3—36no—
189C2×C4×D3—36no—
190C2×C4×D3—36no—
191C2×C4×D3—66no—
192C2×C4×D3—66no—
193C2×C4×D3C4×D316yes—
194C2×C4×D3C4×D316yes—
195C2×C4×D3C22×D316yes—
196C2×C4×D3C22×D316yes—
197C2×Dic6—312no—
198C2×Dic6—312no—
199C3×M16—316no—
200C2×C3×C8—26no—
201C2×C3×C8—36no—
202C2×C3×C8—36no—
203C2×C3×C8—316no—
204C2×C3×C8C22×D316yes—
205C2×C3×C8C22×D316yes—
206#208—316no—
207C2×C3⋊C8—36no—
208C2×C3⋊C8—36no—
209C2×C3⋊C8—316no—
210C2×C3⋊C8—66no—
211C2×C3⋊C8C22×D316yes—
212C2×C3⋊C8C22×D316yes—
213ΓL(1,𝔽25)—316no—
214ΓL(1,𝔽25)—316no—
215ΓL(1,𝔽25)—316no—
216ΓL(1,𝔽25)—616no—
217ΓL(1,𝔽25)C22×D3116yescomplemented
218D3×C8—316no—
219D3×C8—316no—
220D3×C8—316no—
221D3×C8C22×D3116yescomplemented
222C22×C32#13114yescomplemented
223C22×C32C23×C413yes—
224C2×C3⋊D3—212no—
225C2×C3⋊D3C23×C414yes—
226C2×C3×D3—212no—
227C2×C3×D3—212no—
228C2×C3×D3C23×C414yes—
229C2×C3×D3C23×C414yes—
230C32×C4C23×C413yes—
231C32×C4C2511yescharacteristic
232C3⋊Dic3C23×C414yes—
233C3×Dic3C23×C414yes—
234C3×Dic3C23×C414yes—
235C2×#41—912no—
236C2×#41—912no—
237C22×Q8—91no—
238C22×D4—91no—
239C22×D4—91no—
240C22×D4—91no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—91no—
244#131—964no—
245C2×M16—312no—
246C2×M16—912no—
247C2×M16—912no—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8—312no—
251C22×C8—912no—
252C22×C8—912no—
253C22×C8—91no—
254C22×C8D3211yescharacteristic
255C23×C3—21no—
256C23×C3—36no—
257C23×C3—36no—
258C23×C3—64no—
259C23×C3—64no—
260C23×C3C2×C4×D311yescharacteristic
261C23×C3C2×C4×D311yescharacteristic
262C22×D3—24no—
263C22×D3—24no—
264C22×D3—36no—
265C22×D3—36no—
266C22×D3—66no—
267C22×D3—64no—
268C22×D3—64no—
269C22×D3—124no—
270C22×D3C2×C4×D316yes—
271C22×D3C2×C4×D316yes—
272C2×C3×C4—23no—
273C2×C3×C4—23no—
274C2×C3×C4—36no—
275C2×C3×C4—36no—
276C2×C3×C4—312no—
277C2×C3×C4—312no—
278C2×C3×C4—34no—
279C2×C3×C4—34no—
280C2×C3×C4C2×C4×D313yes—
281C2×C3×C4C2×C4×D313yes—
282C2×C3×C4C23×D313yes—
283C2×C3×C4C23×D313yes—
284#67—316no—
285#67—316no—
286#67—316no—
287#67—316no—
288C2×Dic3—36no—
289C2×Dic3—36no—
290C2×Dic3—66no—
291C2×Dic3C2×C4×D316yes—
292C2×Dic3C2×C4×D316yes—
293D12—316no—
294D12—316no—
295C4×D3—312no—
296C4×D3—312no—
297C4×D3—34no—
298C4×D3—34no—
299C4×D3—612no—
300C4×D3—64no—
301C4×D3C2×C4×D3112yes—
302C4×D3C2×C4×D3112yes—
303C4×D3C23×D314yes—
304C4×D3C23×D314yes—
305Dic6—316no—
306Dic6—316no—
307C3×C8—24no—
308C3×C8—34no—
309C3×C8—34no—
310C3×C8C23×D314yes—
311C3×C8C23×D314yes—
312C3⋊C8—34no—
313C3⋊C8—34no—
314C3⋊C8—64no—
315C3⋊C8C23×D314yes—
316C3⋊C8C23×D314yes—
317C2×C32C2×#13116yescomplemented
318C2×C32C24×C411yescharacteristic
319C3⋊D3—24no—
320C3×D3—24no—
321C3×D3—24no—
322C24—31no—
323C24—31no—
324C24—91no—
325#41—916no—
326#41—948no—
327C2×Q8—912no—
328C2×D4—912no—
329C2×D4—912no—
330C2×D4—912no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—36no—
334C22×C4—36no—
335C22×C4—36no—
336C22×C4—36no—
337C22×C4—91no—
338C22×C4—96no—
339C22×C4—96no—
340C22×C4C2×D3211yescharacteristic
341M16—316no—
342M16—916no—
343M16—916no—
344C2×C8—36no—
345C2×C8—36no—
346C2×C8—316no—
347C2×C8—916no—
348C2×C8—916no—
349C2×C8—96no—
350C2×C8C2×D3216yes—
351C22×C3—23no—
352C22×C3—24no—
353C22×C3—34no—
354C22×C3—34no—
355C22×C3—612no—
356C22×C3—612no—
357C22×C3#90114yescomplemented
358C22×C3#90714yescomplemented
359C22×C3C22×C4×D313yes—
360C22×C3C22×C4×D313yes—
361C2×D3—212no—
362C2×D3—212no—
363C2×D3—34no—
364C2×D3—34no—
365C2×D3—64no—
366C2×D3—612no—
367C2×D3—612no—
368C2×D3—1212no—
369C2×D3C22×C4×D314yes—
370C2×D3C22×C4×D314yes—
371C3×C4—23no—
372C3×C4—21no—
373C3×C4—34no—
374C3×C4—34no—
375C3×C4C22×C4×D313yes—
376C3×C4C22×C4×D313yes—
377C3×C4C24×D311yescharacteristic
378C3×C4C24×D311yescharacteristic
379Dic3—34no—
380Dic3—34no—
381Dic3—64no—
382Dic3C22×C4×D314yes—
383Dic3C22×C4×D314yes—
384C32C22×#13111yescharacteristic, complemented, Sylow
385C23—36no—
386C23—36no—
387C23—64no—
388C23—64no—
389C23—96no—
390C23—184no—
391C23C4×D3211yescharacteristic
392Q8—916no—
393D4—916no—
394D4—916no—
395D4—916no—
396C2×C4—36no—
397C2×C4—36no—
398C2×C4—312no—
399C2×C4—312no—
400C2×C4—34no—
401C2×C4—34no—
402C2×C4—96no—
403C2×C4—912no—
404C2×C4—94no—
405C2×C4C4×D3213yes—
406C2×C4C22×D3213yes—
407C8—34no—
408C8—34no—
409C8—94no—
410C8C22×D3214yes—
411C2×C3—21no—
412C2×C3—26no—
413C2×C3—64no—
414C2×C3—64no—
415C2×C3C2×#90116yescomplemented
416C2×C3C2×#90716yescomplemented
417C2×C3C23×C4×D311yescharacteristic
418C2×C3C23×C4×D311yescharacteristic
419D3—24no—
420D3—24no—
421D3—64no—
422D3—64no—
423D3—124no—
424C22—34no—
425C22—34no—
426C22—612no—
427C22—612no—
428C22—94no—
429C22—1812no—
430C22#6380414yesdirect product factor
431C22C2×C4×D3213yes—
432C4—34no—
433C4—34no—
434C4—94no—
435C4C2×C4×D3213yes—
436C4C23×D3211yescharacteristic
437C3—21no—
438C3C22×#90111yescharacteristic, complemented
439C3C22×#90711yescharacteristic, complemented
440C2—64no—
441C2—64no—
442C2—184no—
443C2C2×#6380416yesdirect product factor
444C2C22×C4×D3211yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC2×C32C24×C4
FrattiniC4C23×D32
FittingC22×C32×C8C22
socleC23×C32C22×C4
prime radicalC22×#3798C2
cosocleC32×C4C25
hypercenterC22×C8D32
hypocenterC32C22×#131
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×#3743C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C22×C3×D3×C8C2semidirect
5. C22×#3711C2semidirect
6. C22×#3710C2semidirect
7. C22×D3×C3⋊C8C2semidirect
8. C2×#63804C2direct (N × H)
22. C2×#3743C22semidirect
25. C2×C3×ΓL(1,𝔽25)C22semidirect
26. C2×C3×D3×C8C22semidirect
27. C2×#3711C22semidirect
28. C2×#3710C22semidirect
29. C2×D3×C3⋊C8C22semidirect
30. #63804C22direct (N × H)
43. C22×ΓL(1,𝔽25)D3semidirect
47. C22×D3×C8D3semidirect
69. #3743C23semidirect
72. C3×ΓL(1,𝔽25)C23semidirect
73. C3×D3×C8C23semidirect
74. #3711C23semidirect
75. #3710C23semidirect
76. D3×C3⋊C8C23semidirect
117. C2×ΓL(1,𝔽25)C2×D3semidirect
121. C2×D3×C8C2×D3semidirect
217. ΓL(1,𝔽25)C22×D3semidirect
221. D3×C8C22×D3semidirect
222. C22×C32#131semidirect
317. C2×C32C2×#131semidirect
357. C22×C3#901semidirect
358. C22×C3#907semidirect
384. C32C22×#131semidirect
415. C2×C3C2×#901semidirect
416. C2×C3C2×#907semidirect
430. C22#63804direct (N × H)
438. C3C22×#901semidirect
439. C3C22×#907semidirect
443. C2C2×#63804direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37982
C22×#37432
C22×C3×ΓL(1,𝔽25)2
C22×C3×D3×C82
C22×#37112
C22×#37102
C22×D3×C3⋊C82
C2×#638042
C22×#9073
C22×#9013