Finite Group Databasebeta

← C23×#9049 #19156675

Subgroups — C23×#9049 #19156675

Order 5888.

544 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3100C211yesmaximal, characteristic
2C23×#2932C211yesmaximal, characteristic, complemented
3C23×#1955C211yesmaximal, characteristic, complemented
4C23×#1540C211yesmaximal, characteristic, complemented
5C23×#1486C211yesmaximal, characteristic, complemented
6C23×#1301C211yesmaximal, characteristic, complemented
7C23×#1281C211yesmaximal, characteristic, complemented
8C2×C23×#131—21no—
9C23×#532C2211yescharacteristic
10C2×C4×D4×C23C2211yescharacteristic
11C23×#480C412yes—
12C23×#474—21no—
13C23×#459—21no—
14C23×#447C412yes—
15C4×D8×C23—22no—
16C23×#435—22no—
17C23×#435—22no—
18C23×#431C2211yescharacteristic
19C23×#431C2211yescharacteristic
20C23×#423C2211yescharacteristic
21C2×C23×C4⋊C8C2211yescharacteristic
22C23×#408—21no—
23C2×C23×C22⋊C8—21no—
24C2×C23×C22⋊C8—21no—
25C2×C23×#97C2211yescharacteristic
26C23×#332—21no—
27C23×#331—21no—
28C23×D4⋊C8—22no—
29C2×#41×C23—21no—
30C22×D4×C23C2×C411yescharacteristic
31C23×C4×C23—21no—
32C2×D8×C23—22no—
33C23×#131—42no—
34C23×#131—42no—
35C2×M16×C23—22no—
36C2×M16×C23—21no—
37C22×C8×C23—22no—
38C22×C8×C23—21no—
39C22×C8×C23D411yescharacteristic
40C22×C8×C23D411yescharacteristic
41C23×C4⋊Q8D411yescharacteristic
42C23×#127D411yescharacteristic
43C23×#124—21no—
44C23×#121—21no—
45C23×#121C2×C412yes—
46C4×D4×C23—21no—
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23D412yes—
52C2×C4⋊C4×C23C2×C411yescharacteristic
53C2×C22⋊C4×C23—21no—
54C2×C42×C23C2311yescharacteristic
55C23×#107—22no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8D412yes—
62C23×#102—22no—
63C23×#102—22no—
64C23×C22⋊C8—24no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8—42no—
67C23×C22⋊C8—42no—
68C23×C22⋊C8C2×C412yes—
69C23×#97—21no—
70C23×#97—21no—
71C4×C8×C23—21no—
72C4×C8×C23—21no—
73C24×C23—21no—
74#41×C23—41no—
75#41×C23—41no—
76#41×C23—41no—
77#41×C23—41no—
78C2×Q8×C23—21no—
79C2×D4×C23—22no—
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—21no—
84C2×D4×C23—42no—
85C2×D4×C23C22⋊C412yes—
86C22×C4×C23—22no—
87C22×C4×C23—21no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23C22×C411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96C22×C4×C23#4111yescharacteristic
97D8×C23—44no—
98M16×C23—44no—
99M16×C23—41no—
100M16×C23—41no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—22no—
104C2×C8×C23—22no—
105C2×C8×C23—21no—
106C2×C8×C23—21no—
107C2×C8×C23—21no—
108C2×C8×C23—44no—
109C2×C8×C23—41no—
110C2×C8×C23—41no—
111C2×C8×C23—41no—
112C2×C8×C23—41no—
113C2×C8×C23D812yescomplemented
114C4⋊C4×C23—21no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23C22⋊C412yes—
118C22⋊C4×C23—22no—
119C22⋊C4×C23—22no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C42×C23C2×D411yescharacteristic
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23#4111yescharacteristic
126#9049C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—22no—
128C23×C23—22no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23C4×D411yescharacteristic
136Q8×C23—41no—
137Q8×C23—41no—
138D4×C23—22no—
139D4×C23—22no—
140D4×C23—42no—
141D4×C23—41no—
142D4×C23—41no—
143D4×C23—42no—
144D4×C23—42no—
145D4×C23—41no—
146D4×C23—41no—
147D4×C23—81no—
148C2×C4×C23—22no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—81no—
157C2×C4×C23—81no—
158C2×C4×C23—81no—
159C2×C4×C23—81no—
160C2×C4×C23C2×C22⋊C411yescharacteristic
161C2×C4×C23C4×D411yescharacteristic
162C2×C4×C23#12011yescharacteristic
163C2×C4×C23#12111yescharacteristic
164C2×C4×C23#12111yescharacteristic
165C2×C4×C23#12311yescharacteristic
166C2×C4×C23C2×D811yescharacteristic
167C2×C4×C23#13511yescharacteristic
168C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
170C8×C23—22no—
171C8×C23—42no—
172C8×C23—42no—
173C8×C23—41no—
174C8×C23—81no—
175C8×C23—81no—
176#3100C2×C2311yescharacteristic
177#2932C2×C2311yescharacteristic, complemented
178#1955C2×C2311yescharacteristic, complemented
179#1540C2×C2311yescharacteristic, complemented
180#1486C2×C2311yescharacteristic, complemented
181#1301C2×C2311yescharacteristic, complemented
182#1281C2×C2311yescharacteristic, complemented
183C22×C23—21no—
184C22×C23—21no—
185C22×C23—42no—
186C22×C23—42no—
187C22×C23—42no—
188C22×C23—42no—
189C22×C23—42no—
190C22×C23—42no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23#38611yescharacteristic
196C22×C23C4×D811yescharacteristic
197C22×C23#44211yescharacteristic
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—21no—
201C4×C23—21no—
202C4×C23—42no—
203C4×C23—81no—
204C4×C23#44711yescharacteristic
205C4×C23#44911yescharacteristic
206C4×C23#45911yescharacteristic
207C4×C23#46311yescharacteristic
208C2×#131—21no—
209#532C22×C2311yescharacteristic
210C2×C4×D4C22×C2311yescharacteristic
211#480C4×C2312yes—
212#474—21no—
213#459—21no—
214#447C4×C2312yes—
215C4×D8—22no—
216#435—22no—
217#435—22no—
218#431C22×C2311yescharacteristic
219#431C22×C2311yescharacteristic
220#423C22×C2311yescharacteristic
221C2×C4⋊C8C22×C2311yescharacteristic
222#408—21no—
223C2×C22⋊C8—21no—
224C2×C22⋊C8—21no—
225C2×#97C22×C2311yescharacteristic
226#332—21no—
227#331—21no—
228D4⋊C8—22no—
229C2×C23—21no—
230C2×C23—21no—
231C2×C23—82no—
232C2×C23—82no—
233C2×C23#187911yescharacteristic
234C2×C23#188111yescharacteristic
235C2×C23#189111yescharacteristic
236C2×#41—21no—
237C22×D4C2×C4×C2311yescharacteristic
238C23×C4—21no—
239C2×D8—22no—
240#131—42no—
241#131—42no—
242C2×M16—22no—
243C2×M16—21no—
244C22×C8—22no—
245C22×C8—21no—
246C22×C8D4×C2311yescharacteristic
247C22×C8D4×C2311yescharacteristic
248C4⋊Q8D4×C2311yescharacteristic
249#127D4×C2311yescharacteristic
250#124—21no—
251#121—21no—
252#121C2×C4×C2312yes—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4D4×C2312yes—
259C2×C4⋊C4C2×C4×C2311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C42C23×C2311yescharacteristic
262#107—22no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8D4×C2312yes—
269#102—22no—
270#102—22no—
271C22⋊C8—24no—
272C22⋊C8—21no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8C2×C4×C2312yes—
276#97—21no—
277#97—21no—
278C4×C8—21no—
279C4×C8—21no—
280C23#904911yescharacteristic, direct product factor, Sylow
281C24—21no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4C22⋊C4×C2312yes—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C22×C4×C2311yescharacteristic
303C22×C4C2×D4×C2311yescharacteristic
304C22×C4#41×C2311yescharacteristic
305D8—44no—
306M16—44no—
307M16—41no—
308M16—41no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—44no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8D8×C2312yescomplemented
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4C22⋊C4×C2312yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42C2×D4×C2311yescharacteristic
331C42C2×D4×C2311yescharacteristic
332C42C2×D4×C2311yescharacteristic
333C42#41×C2311yescharacteristic
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23C4×D4×C2311yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C2×C22⋊C4×C2311yescharacteristic
368C2×C4C4×D4×C2311yescharacteristic
369C2×C4C23×#12011yescharacteristic
370C2×C4C23×#12111yescharacteristic
371C2×C4C23×#12111yescharacteristic
372C2×C4C23×#12311yescharacteristic
373C2×C4C2×D8×C2311yescharacteristic
374C2×C4C23×#13511yescharacteristic
375C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
377C8—22no—
378C8—42no—
379C8—42no—
380C8—41no—
381C8—81no—
382C8—81no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C23×#38611yescharacteristic
396C22C4×D8×C2311yescharacteristic
397C22C23×#44211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4C23×#44711yescharacteristic
405C4C23×#44911yescharacteristic
406C4C23×#45911yescharacteristic
407C4C23×#46311yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2—82no—
412C2C23×#187911yescharacteristic
413C2C23×#188111yescharacteristic
414C2C23×#189111yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#3100C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2932C2semidirect
3. C23×#1955C2semidirect
4. C23×#1540C2semidirect
5. C23×#1486C2semidirect
6. C23×#1301C2semidirect
7. C23×#1281C2semidirect
113. C2×C8×C23D8semidirect
126. #9049C23direct (N × H)
177. #2932C2×C23semidirect
178. #1955C2×C23semidirect
179. #1540C2×C23semidirect
180. #1486C2×C23semidirect
181. #1301C2×C23semidirect
182. #1281C2×C23semidirect
280. C23#9049direct (N × H)
321. C2×C8D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31002
C23×#29322
C23×#19552
C23×#15402
C23×#14862
C23×#13012
C23×#12812
#904923