Finite Group Databasebeta

← C23×#9299 #19156925

Subgroups — C23×#9299 #19156925

Order 5888.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3205C211yesmaximal, characteristic
2C23×#2328C211yesmaximal, characteristic, complemented
3C23×#1963C211yesmaximal, characteristic, complemented
4C23×#1545C211yesmaximal, characteristic, complemented
5C23×#1517C211yesmaximal, characteristic, complemented
6C23×#1286C211yesmaximal, characteristic, complemented
7C23×#1274C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C23×#127C2211yescharacteristic
10C2×C4×D4×C23—21no—
11C23×#493C412yescomplemented
12C23×#486C412yescomplemented
13C23×#459—22no—
14C23×#447—22no—
15C23×#431C2211yescharacteristic
16C23×#431C2211yescharacteristic
17C23×#423C2211yescharacteristic
18C2×C23×#104—21no—
19C2×C23×#104—21no—
20C23×#390—21no—
21C23×#386—22no—
22C23×#386—22no—
23C23×#382C2211yescharacteristic
24C23×#376C2211yescharacteristic
25C23×#331—22no—
26C23×#327—22no—
27C2×#41×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23—21no—
30C22×D4×C23D412yes—
31C23×C4×C23—22no—
32C2×D8×C23—24no—
33C2×M16×C23—21no—
34C23×C4⋊Q8C2×C411yescharacteristic, complemented
35C23×#127—21no—
36C23×#127—21no—
37C23×#127C2×C412yes—
38C23×#127C2×C411yescharacteristic, complemented
39C23×#127D412yes—
40C23×#124—21no—
41C23×#121—21no—
42C23×#121—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23—42no—
46C4×D4×C23—42no—
47C2×C4⋊C4×C23—21no—
48C2×C22⋊C4×C23—22no—
49C2×C22⋊C4×C23—22no—
50C2×C22⋊C4×C23—22no—
51C2×C42×C23D411yescharacteristic
52C2×C42×C23D411yescharacteristic
53C2×C42×C23C2311yescharacteristic
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×#104—42no—
59C23×#104—42no—
60C23×#102—24no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C4×C8×C23—21no—
64C4×C8×C23C2×C412yescomplemented
65C23×#95—21no—
66C23×#95—21no—
67C23×#95—21no—
68C23×#95D412yes—
69C24×C23—22no—
70#41×C23—41no—
71#41×C23—41no—
72#41×C23—41no—
73#41×C23—41no—
74C2×Q8×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—22no—
77C2×D4×C23—21no—
78C2×D4×C23—22no—
79C2×D4×C23—21no—
80C2×D4×C23—22no—
81C2×D4×C23—22no—
82C2×D4×C23—22no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—82no—
89C2×D4×C23—82no—
90C2×D4×C23C22⋊C414yes—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—22no—
94C22×C4×C23—44no—
95C22×C4×C23—41no—
96C22×C4×C23—82no—
97C22×C4×C23—82no—
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23C2×D411yescharacteristic
101D8×C23—84no—
102M16×C23—41no—
103M16×C23—41no—
104C2×C8×C23—22no—
105C2×C8×C23—41no—
106C2×C8×C23—41no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C22⋊C4×C23—44no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—82no—
115C22⋊C4×C23—82no—
116C42×C23—21no—
117C42×C23—21no—
118C42×C23—21no—
119C42×C23D812yescomplemented
120C42×C23C22×C411yescharacteristic
121C42×C23C2×D411yescharacteristic
122C42×C23#4111yescharacteristic
123C42×C23#4111yescharacteristic
124#9299C2311yesmaximal, characteristic, direct product factor, Sylow
125C23×C23—24no—
126C23×C23—42no—
127C23×C23—42no—
128C23×C23—42no—
129C23×C23—42no—
130C23×C23—42no—
131C23×C23—42no—
132C23×C23—41no—
133C23×C23#12011yescharacteristic
134Q8×C23—41no—
135Q8×C23—41no—
136D4×C23—44no—
137D4×C23—42no—
138D4×C23—42no—
139D4×C23—41no—
140D4×C23—42no—
141D4×C23—41no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23—82no—
145D4×C23—82no—
146D4×C23—82no—
147D4×C23—81no—
148D4×C23—81no—
149D4×C23—81no—
150C2×C4×C23—22no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—42no—
158C2×C4×C23—42no—
159C2×C4×C23—41no—
160C2×C4×C23—42no—
161C2×C4×C23—41no—
162C2×C4×C23—88no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23C2×C22⋊C411yescharacteristic
166C2×C4×C23C4×D411yescharacteristic
167C2×C4×C23C4×D411yescharacteristic
168C2×C4×C23#12111yescharacteristic
169C2×C4×C23#12111yescharacteristic
170C2×C4×C23#12311yescharacteristic
171C2×C4×C23C2×D811yescharacteristic
172C2×C4×C23#13511yescharacteristic
173C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
175C8×C23—42no—
176C8×C23—81no—
177C8×C23—81no—
178#3205C2×C2311yescharacteristic
179#2328C2×C2311yescharacteristic, complemented
180#1963C2×C2311yescharacteristic, complemented
181#1545C2×C2311yescharacteristic, complemented
182#1517C2×C2311yescharacteristic, complemented
183#1286C2×C2311yescharacteristic, complemented
184#1274C2×C2311yescharacteristic, complemented
185C22×C23—21no—
186C22×C23—21no—
187C22×C23—44no—
188C22×C23—44no—
189C22×C23—44no—
190C22×C23—82no—
191C22×C23—82no—
192C22×C23—82no—
193C22×C23—82no—
194C22×C23—81no—
195C22×C23—81no—
196C22×C23—81no—
197C22×C23#38611yescharacteristic
198C22×C23#44711yescharacteristic
199C22×C23#44911yescharacteristic
200C4×C23—21no—
201C4×C23—21no—
202C4×C23—21no—
203C4×C23—21no—
204C4×C23—42no—
205C4×C23—81no—
206C4×C23—82no—
207C4×C23—81no—
208C4×C23C4×D811yescharacteristic
209C4×C23#44211yescharacteristic
210C4×C23#45911yescharacteristic
211C4×C23#46311yescharacteristic
212#532C22×C2311yescharacteristic
213C2×#127C22×C2311yescharacteristic
214C2×C4×D4—21no—
215#493C4×C2312yescomplemented
216#486C4×C2312yescomplemented
217#459—22no—
218#447—22no—
219#431C22×C2311yescharacteristic
220#431C22×C2311yescharacteristic
221#423C22×C2311yescharacteristic
222C2×#104—21no—
223C2×#104—21no—
224#390—21no—
225#386—22no—
226#386—22no—
227#382C22×C2311yescharacteristic
228#376C22×C2311yescharacteristic
229#331—22no—
230#327—22no—
231C2×C23—21no—
232C2×C23—21no—
233C2×C23—84no—
234C2×C23—161no—
235C2×C23#187911yescharacteristic
236C2×C23#190411yescharacteristic
237C2×C23#191111yescharacteristic
238C2×#41—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4D4×C2312yes—
242C23×C4—22no—
243C2×D8—24no—
244C2×M16—21no—
245C4⋊Q8C2×C4×C2311yescharacteristic, complemented
246#127—21no—
247#127—21no—
248#127C2×C4×C2312yes—
249#127C2×C4×C2311yescharacteristic, complemented
250#127D4×C2312yes—
251#124—21no—
252#121—21no—
253#121—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—42no—
257C4×D4—42no—
258C2×C4⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42D4×C2311yescharacteristic
263C2×C42D4×C2311yescharacteristic
264C2×C42C23×C2311yescharacteristic
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269#104—42no—
270#104—42no—
271#102—24no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C4×C8—21no—
275C4×C8C2×C4×C2312yescomplemented
276#95—21no—
277#95—21no—
278#95—21no—
279#95D4×C2312yes—
280C23#929911yescharacteristic, direct product factor, Sylow
281C24—22no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4C22⋊C4×C2314yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4C2×D4×C2311yescharacteristic
311C22×C4C2×D4×C2311yescharacteristic
312C22×C4C2×D4×C2311yescharacteristic
313D8—84no—
314M16—41no—
315M16—41no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—44no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42D8×C2312yescomplemented
332C42C22×C4×C2311yescharacteristic
333C42C2×D4×C2311yescharacteristic
334C42#41×C2311yescharacteristic
335C42#41×C2311yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23C23×#12011yescharacteristic
345Q8—41no—
346Q8—41no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—42no—
352D4—41no—
353D4—42no—
354D4—42no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—88no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4C2×C22⋊C4×C2311yescharacteristic
377C2×C4C4×D4×C2311yescharacteristic
378C2×C4C4×D4×C2311yescharacteristic
379C2×C4C23×#12111yescharacteristic
380C2×C4C23×#12111yescharacteristic
381C2×C4C23×#12311yescharacteristic
382C2×C4C2×D8×C2311yescharacteristic
383C2×C4C23×#13511yescharacteristic
384C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
385C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
386C8—42no—
387C8—81no—
388C8—81no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C23×#38611yescharacteristic
402C22C23×#44711yescharacteristic
403C22C23×#44911yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4C4×D8×C2311yescharacteristic
413C4C23×#44211yescharacteristic
414C4C23×#45911yescharacteristic
415C4C23×#46311yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—161no—
420C2C23×#187911yescharacteristic
421C2C23×#190411yescharacteristic
422C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3205C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2328C2semidirect
3. C23×#1963C2semidirect
4. C23×#1545C2semidirect
5. C23×#1517C2semidirect
6. C23×#1286C2semidirect
7. C23×#1274C2semidirect
11. C23×#493C4semidirect
12. C23×#486C4semidirect
34. C23×C4⋊Q8C2×C4semidirect
38. C23×#127C2×C4semidirect
64. C4×C8×C23C2×C4semidirect
119. C42×C23D8semidirect
124. #9299C23direct (N × H)
179. #2328C2×C23semidirect
180. #1963C2×C23semidirect
181. #1545C2×C23semidirect
182. #1517C2×C23semidirect
183. #1286C2×C23semidirect
184. #1274C2×C23semidirect
215. #493C4×C23semidirect
216. #486C4×C23semidirect
245. C4⋊Q8C2×C4×C23semidirect
249. #127C2×C4×C23semidirect
275. C4×C8C2×C4×C23semidirect
280. C23#9299direct (N × H)
331. C42D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32052
C23×#23282
C23×#19632
C23×#15452
C23×#15172
C23×#12862
C23×#12742
#929923