Finite Group Databasebeta

← C23×#8349 #19155975

Subgroups — C23×#8349 #19155975

Order 5888.

720 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3493C211yesmaximal, characteristic
2C23×#1750C211yesmaximal, characteristic, complemented
3C23×#1532C212yesmaximal, complemented
4C23×#1494C212yesmaximal, complemented
5C23×#1464C211yesmaximal, characteristic, complemented
6C23×#556—22no—
7C23×#555—22no—
8C23×#553—22no—
9C23×#552—22no—
10C23×#544C411yescharacteristic
11C23×#538—21no—
12C23×#538C412yes—
13C23×#537C411yescharacteristic
14C23×#529—21no—
15C23×#529C2212yes—
16C23×#525—21no—
17C23×#520—21no—
18C2×C23×#117C2211yescharacteristic
19C23×#432C2212yes—
20C2×C23×C22⋊C8—21no—
21C2×C23×#97C2211yescharacteristic
22C2×C4×C8×C23C2211yescharacteristic
23C23×#336—21no—
24C23×#330—22no—
25C23×#329—22no—
26C23×Q8⋊C8—22no—
27C23×D4⋊C8—22no—
28C23×#324—21no—
29C23×#323—21no—
30C2×#41×C23—21no—
31C23×C4×C23D411yescharacteristic
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C23×C4⋊Q8—22no—
35C23×#126—22no—
36C23×#126—22no—
37C23×#126—22no—
38C23×#126—22no—
39C23×#125—21no—
40C23×#125—22no—
41C23×#125D412yes—
42C23×#124—21no—
43C23×#124—22no—
44C23×#124—22no—
45C23×#124D412yes—
46C23×#123—22no—
47C23×#123—22no—
48C23×#123—22no—
49C23×C22⋊Q8—21no—
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8—21no—
52C23×C22⋊Q8—22no—
53C23×C22⋊Q8C2×C412yes—
54C23×#121—21no—
55C23×#121—22no—
56C23×#121—21no—
57C23×#121—22no—
58C23×#121—22no—
59C23×#121—21no—
60C23×#121C2×C412yes—
61C23×#120—22no—
62C4×Q8×C23—22no—
63C4×Q8×C23—21no—
64C4×Q8×C23D412yes—
65C4×D4×C23—22no—
66C4×D4×C23—21no—
67C4×D4×C23—22no—
68C4×D4×C23—22no—
69C4×D4×C23—22no—
70C4×D4×C23—22no—
71C4×D4×C23D412yes—
72C23×#117—22no—
73C23×#117—22no—
74C23×#117—21no—
75C23×#117—22no—
76C23×#117—22no—
77C23×#117—22no—
78C2×C4⋊C4×C23—22no—
79C2×C4⋊C4×C23C2×C411yescharacteristic
80C2×C4⋊C4×C23D411yescharacteristic
81C2×C22⋊C4×C23C2×C411yescharacteristic
82C2×C22⋊C4×C23D411yescharacteristic
83C2×C42×C23D411yescharacteristic
84C2×C42×C23C2311yescharacteristic
85C23×C4⋊C8—22no—
86C23×C22⋊C8—24no—
87C23×C22⋊C8—22no—
88C23×#97—22no—
89C23×#97—22no—
90C4×C8×C23—22no—
91C24×C23C22⋊C411yescharacteristic
92#41×C23—42no—
93#41×C23—42no—
94C2×Q8×C23—21no—
95C2×Q8×C23—21no—
96C2×Q8×C23C22⋊C412yes—
97C2×D4×C23—21no—
98C2×D4×C23—22no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—21no—
102C2×D4×C23C22⋊C412yes—
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23—21no—
107C22×C4×C23—21no—
108C22×C4×C23—22no—
109C22×C4×C23—41no—
110C22×C4×C23C22⋊C411yescharacteristic
111C22×C4×C23C22⋊C411yescharacteristic
112C22×C4×C23C22⋊C411yescharacteristic
113C22×C4×C23C22×C411yescharacteristic
114C22×C4×C23C2×D411yescharacteristic
115C22×C4×C23C2×D411yescharacteristic
116C2×C8×C23—22no—
117C2×C8×C23—22no—
118C2×C8×C23—41no—
119C2×C8×C23—41no—
120C2×C8×C23—41no—
121C2×C8×C23—41no—
122C2×C8×C23—42no—
123C4⋊C4×C23—21no—
124C4⋊C4×C23—22no—
125C4⋊C4×C23—21no—
126C4⋊C4×C23—22no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—22no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23C22⋊C412yes—
136C4⋊C4×C23C22⋊C412yes—
137C22⋊C4×C23—22no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—22no—
142C22⋊C4×C23—22no—
143C22⋊C4×C23—22no—
144C22⋊C4×C23—22no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C42×C23—22no—
149C42×C23—22no—
150C42×C23—21no—
151C42×C23—21no—
152C42×C23—22no—
153C42×C23C2×D412yes—
154C42×C23C2×D412yes—
155#8349C2311yesmaximal, characteristic, direct product factor, Sylow
156C23×C23—21no—
157C23×C23—22no—
158C23×C23—21no—
159C23×C23—21no—
160C23×C23—41no—
161C23×C23—41no—
162C23×C23C2×C22⋊C411yescharacteristic
163Q8×C23—42no—
164Q8×C23—42no—
165Q8×C23—81no—
166D4×C23—42no—
167D4×C23—42no—
168D4×C23—42no—
169D4×C23—42no—
170D4×C23—82no—
171D4×C23—81no—
172D4×C23—82no—
173C2×C4×C23—21no—
174C2×C4×C23—21no—
175C2×C4×C23—21no—
176C2×C4×C23—21no—
177C2×C4×C23—21no—
178C2×C4×C23—22no—
179C2×C4×C23—21no—
180C2×C4×C23—22no—
181C2×C4×C23—22no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—44no—
187C2×C4×C23—41no—
188C2×C4×C23—41no—
189C2×C4×C23—82no—
190C2×C4×C23—82no—
191C2×C4×C23#10412yes—
192C2×C4×C23C2×C22⋊C412yes—
193C2×C4×C23#12012yes—
194C2×C4×C23#12011yescharacteristic
195C2×C4×C23#12011yescharacteristic
196C8×C23—42no—
197C8×C23—42no—
198C8×C23—82no—
199#3493C2×C2311yescharacteristic
200#1750C2×C2311yescharacteristic, complemented
201#1532C2×C2312yescomplemented
202#1494C2×C2312yescomplemented
203#1464C2×C2311yescharacteristic, complemented
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—42no—
207C22×C23—42no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—42no—
211C22×C23—42no—
212C22×C23—41no—
213C22×C23—81no—
214C22×C23#37911yescharacteristic
215C22×C23C2×#10411yescharacteristic
216C22×C23#42111yescharacteristic
217C4×C23—22no—
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—22no—
221C4×C23—22no—
222C4×C23—41no—
223C4×C23—42no—
224C4×C23—42no—
225C4×C23—41no—
226C4×C23—42no—
227#556—22no—
228#555—22no—
229#553—22no—
230#552—22no—
231#544C4×C2311yescharacteristic
232#538—21no—
233#538C4×C2312yes—
234#537C4×C2311yescharacteristic
235#529—21no—
236#529C22×C2312yes—
237#525—21no—
238#520—21no—
239C2×#117C22×C2311yescharacteristic
240#432C22×C2312yes—
241C2×C22⋊C8—21no—
242C2×#97C22×C2311yescharacteristic
243C2×C4×C8C22×C2311yescharacteristic
244#336—21no—
245#330—22no—
246#329—22no—
247Q8⋊C8—22no—
248D4⋊C8—22no—
249#324—21no—
250#323—21no—
251C2×C23—21no—
252C2×C23—21no—
253C2×C23—82no—
254C2×C23—81no—
255C2×C23#178911yescharacteristic
256C2×C23#179111yescharacteristic
257C2×C23#179411yescharacteristic
258C2×#41—21no—
259C23×C4D4×C2311yescharacteristic
260C22×C8—21no—
261C22×C8—21no—
262C4⋊Q8—22no—
263#126—22no—
264#126—22no—
265#126—22no—
266#126—22no—
267#125—21no—
268#125—22no—
269#125D4×C2312yes—
270#124—21no—
271#124—22no—
272#124—22no—
273#124D4×C2312yes—
274#123—22no—
275#123—22no—
276#123—22no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—22no—
281C22⋊Q8C2×C4×C2312yes—
282#121—21no—
283#121—21no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121—21no—
288#121C2×C4×C2312yes—
289#120—22no—
290C4×Q8—22no—
291C4×Q8—21no—
292C4×Q8D4×C2312yes—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4D4×C2312yes—
300#117—22no—
301#117—22no—
302#117—21no—
303#117—22no—
304#117—22no—
305#117—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×C4×C2311yescharacteristic
308C2×C4⋊C4D4×C2311yescharacteristic
309C2×C22⋊C4C2×C4×C2311yescharacteristic
310C2×C22⋊C4D4×C2311yescharacteristic
311C2×C42D4×C2311yescharacteristic
312C2×C42C23×C2311yescharacteristic
313C4⋊C8—22no—
314C22⋊C8—24no—
315C22⋊C8—22no—
316#97—22no—
317#97—22no—
318C4×C8—22no—
319C23#834911yescharacteristic, direct product factor, Sylow
320C24C22⋊C4×C2311yescharacteristic
321#41—42no—
322#41—42no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8C22⋊C4×C2312yes—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4C22⋊C4×C2312yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4C22⋊C4×C2311yescharacteristic
340C22×C4C22⋊C4×C2311yescharacteristic
341C22×C4C22⋊C4×C2311yescharacteristic
342C22×C4C22×C4×C2311yescharacteristic
343C22×C4C2×D4×C2311yescharacteristic
344C22×C4C2×D4×C2311yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4C22⋊C4×C2312yes—
365C4⋊C4C22⋊C4×C2312yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—21no—
381C42—22no—
382C42C2×D4×C2312yes—
383C42C2×D4×C2312yes—
384C23—21no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—41no—
389C23—41no—
390C23C2×C22⋊C4×C2311yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—81no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—81no—
399D4—82no—
400D4—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4C23×#10412yes—
420C2×C4C2×C22⋊C4×C2312yes—
421C2×C4C23×#12011yescharacteristic
422C2×C4C23×#12011yescharacteristic
423C2×C4C23×#12012yes—
424C8—42no—
425C8—42no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—81no—
437C22C23×#37911yescharacteristic
438C22C2×C23×#10411yescharacteristic
439C22C23×#42111yescharacteristic
440C4—21no—
441C4—21no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—42no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—82no—
454C2C23×#178911yescharacteristic
455C2C23×#179111yescharacteristic
456C2C23×#179411yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3493C2
LCSG ▷ C22×C4 ▷ 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×#1750C2semidirect
3. C23×#1532C2semidirect
4. C23×#1494C2semidirect
5. C23×#1464C2semidirect
155. #8349C23direct (N × H)
200. #1750C2×C23semidirect
201. #1532C2×C23semidirect
202. #1494C2×C23semidirect
203. #1464C2×C23semidirect
319. C23#8349direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34932
C23×#17502
C23×#15322
C23×#14942
C23×#14642
#834923